So today we're going to talk about randomness and probability and sampling and sampling distribution. So this is actually a very important notebook that we're going to work our way through. We're going to start off with by rolling some dice and flipping some coins so that you can understand this idea behind probability and how that is eventually the randomness in that it's eventually going to help us in data science to analyze the data that we work with. So we're here in notebook number seven randomness and sampling. So let's just have a look at the packages that we're going to use. Remember I use that line of code if I work on a MacBook. So in this instance I'm recording on a Windows machine so that's not going to be too important but we'll run that line of code anyway and then my usual %load underscore ext and then the google.colarep.data underscore table that's my magic command there just so that the tables print out nicely. And then things we've seen before so we're going to import from numpy or at least import numpy with a namespace abbreviation np we're going to import from scipy the stats module we're going to import pandas use the namespace abbreviation pd so let's do all of those and then for plotting we're going to do graph objects we're going to do io just to set the plotting underscore white theme for us there plot the express and then also figure underscore factory that's a different module and we're going to import that as ff and let's just set that default to plotly white. So we're going to talk a little bit about randomness you flip a coin you roll a die random numbers come up and a computer can to some extent simulate that because as human beings we're very very bad if i asked you to just say to say from zero to nine give me a list of a hundred random numbers or do it a hundred times so nine eight seven six four three there's no ways that human beings come up with real random values we form little patterns in our head so in somehow we've got to replicate that on a computer and what a computer does it uses an algorithm and it can use infinitesimal little time fractions on the clock cycles of your computer cpu there's all sorts of algorithms that it can use to come up with these random these numbers but they're not really random in as much as we won't go into to the depths of of this topic but they are actually pseudo random numbers because they come from some algorithm so we've imported numpy as np and we can do this seeding so numpy dot random dot seed so that seed function gives a value to the algorithm so that when it generates random numbers and you run the cell again i run it you run it on your computer we're going to get the same set of pseudo random numbers and we do that a lot in data science if we you know if we involve with teaching or if we want to share our code and we put those random seeds in there because we want those results to be reproducible so if we're putting in any integer value there i've put in 42 if you wrote this line of code and you put in 42 you're going to get the same set of pseudo random numbers as me and then in this random module there's also a choice function and what you do with a choice function is you pass a python list and this time i'm passing a list with two elements and they're both strings and what is going to happen is one of those two are going to be chosen at random and as i said i'm you know what the result is going to be when i do 42 because i've run this line of code before and if you do exactly the same you also going to get the same result so let's look randomly or see the randomly what the value is that was chosen and we can see that's heads so heads came out at random so this choice function also takes a second argument so i've set the comma 10 and this time it's going to do this 10 times over so it's going to draw heads or tails for me 10 times over but i've set the random c to 42 again so that if me and you if we both are in this line of code we're going to get the same pseudo random numbers and this time it's heads tails heads heads tails heads heads heads tails as simple as simple as that so let's ram things up a bit we you know this is a powerful computer we're running this in the cloud on google servers quite powerful machines obviously so let's go nuts so i'm going to this time set the c to none in other words now i'm going to get real pseudo random numbers in other ways if i run this line of code twice i'm going to get different values if you run it you're going to get different values so i'm going to create this computer variable called flips and to that i assign a panel's data frame so pd dot data frame and what i'm going to do is generate a column through just using index notation so a dictionary there so you see the curly braces curly braces so it's a dictionary so key value pairs my key is going to be flip so that becomes the name of the column and the values is numpy dot random duck choice heads or tails 10 000 times and let's look at the first five rows with the head method so that's going to run in it got heads heads tails tails heads as i say you run it you might get something different i run it again i'm going to get something different but it's 10 000 so the random choice function random duck choice function that's an equal likelihood for both so let's just see what happened in this 10 000 we can do because this is a data frame so we can call the series or just that column by saying flips dot flip because remember i made flip the column header and dot value underscore counts so that's going to tell me you know the frequency of the sample space element so it got 5000 and 97 heads and 4903 tails that's really close to each other it's theoretically we're going to get a 50 50 but we'll have to do a lot more than 10 000 you know to get closer to that theoretical 50 50 and there's actually a term for that and we'll get to that a little bit later and that's that how we get from our actual values to the theoretical values there's a little term for that so that's the coin flip let's roll a die a fair die so by fair we usually mean all the six values have equal likelihood of landing face up that's not some loaded die that's going to fall on certain values more frequently than others so we can simulate that as well and i'm going to create another pandas data frame and i'm going to call it die and to that i'm going to assign again a single column and i'm setting that up right there as a dictionary key value pair so die is my column header my values are going to be a random dot choice and the choice is going to be made from this python list that i've created up here sides and it's just a python list of six values one two three four five six and it's going to do that 10 000 times so let's do that so we've rolled our die within a fraction of a second we've rolled the 10 000 times and we can just create a bar chart so i'm going to do that from the plotly express module so px dot bar and then we want to know what the variable is we're going to do the bar chart of so x equals the sides so what are the sides remember if i hover over there we'll actually see the tooltip come up it's one two three four five six the six items on the y axis i want the count for each but just be very careful there remember what this value counts is going to do it's going to put the most frequent one at the top the mode at the top but we wanted to sort by the index and the index is one two three four five six so that you know the first bar is the value counts for the four for one and the second bar is the value counts for two so we do put this dot sort index method there and then from that we just want the values and we want it converted to a list i think you know by now you know what this little line of code does you can just tease it out as it goes along i'm going to put a title there frequency of faces falling 10 000 rolls and instead of x you know i'm changing the labels because it's going to put x on the x axis i want face values and instead of y i want frequency so let's have a look at what this looks like and you can see there now this is what we would term a uniform distribution and we'll talk about distributions a little bit later but you can see all the values are almost equal equal number of times they landed face up so this is you'll see later this is an empirical result the result from our actual data theoretically it should be a six a six a six etc because each value has one over six chance of landing face up so let's talk a little bit about probabilities because it becomes very important data science as we work as we work and it's not easy at times the first time you see this but if you sit down after the fact after having worked through it then it starts making sense so the first thing in probability theory and we use it it's a branch of mathematics very important branch of mathematics where we just investigate random events and it's all based on set theory so well most of mathematics all of mathematics the axiomatic system of mathematics is based on set theory and so you can really take very long and complex courses at university about set theory and what we're just going to view a set as as a collection of objects and you know we're using very poor language our language skills and our communal understanding of what the definitions are in a dictionary of these terms that we're using but let's just use that conceptualization is that a word of a collection of elements and those can be numbers or can be many things just this collection and that that is a set and that element in a set we would say it is a member of that set so there's some terms that we use and here terms that you've seen before intersection and union remember the intersection of two think of Venn diagrams remember those where they intersect in other words we're looking for members that are common to two sets so that member occurs in set number one or set number A set A and it occurs in set B and if that is a member of both it lives in the intersection of those two sets if I combine those two sets together then I have the union of those and remember where the union the little bit that's the intersection of a union those elements actually get duplicated because they occur in both sets that's why they're in the intersection so we just have to delete those duplicates and later that's going to become important as well but I think if you just think it through a little bit if there is an intersection between two sets of course the elements that are in the intersection are duplicated when we do the union and we don't want duplications in the union so it's just the elements themselves and then we have this concept of a universal set so imagine you're doing some study you're working in a lab or working with clients doesn't matter what it is and you're collecting data for specific variable there's a range of values for that variable but you're only working with 10 samples so not all the possible values that are in that interval that are possible for that variable is going to occur in your data set but that largest set we usually refer to that as the universal set and our sample space comes from that or sometimes is that universal set the sample space for variable in any case if not all the values occur in our little sample that are in the universal set the ones that didn't occur we call that the complement so those are the values that are that are in the universal set but that did not occur in our little sample so that would be the complement so terms you'll come across also terms like experiment event and outcome and sometimes they use very interchangeably but they are you know slightly different things I think if we can start with an experiment that's just the thing you do that's the thing that you set up and imagine it is an organism in a laboratory and you're going to take some fluorescent score from it you know that's your experiment you take you actively doing something so that you can get some result out of it that you can jot down or capture in your spreadsheet or in your database that's your experiment and you do that over and over again and then the event is that actual value that you're going to collect from that experiment there's a value but then there's also an outcome of of events and we'll see that when we talk about this flipping our our fair coin here twice so independent events means that if I do this twice the second one the first one has no bearing on what happens to the second one so that second one doesn't care what happened to the first one if you flip a coin twice the coin doesn't care what the previous flip was that's every time you flip that coin there's an independent event but now we change our experiment and we say my experiment is flipping the coin twice each time that I flip the coin there's an event but my outcome is jotting down both of the events so I actually have four outcomes if you think about it I can toss a flip the coin head head head tail tail head or tail tail so I can certainly do that and my outcome is now not just the single event but that combination of events I want my outcome is I'm throwing I'm flipping the coin twice so that event the what I'm recording heads or tails that happens twice so each of these four events if it's a fair coin they have an equal likelihood of occurring so flipping a head head would occur 0.25 or 25 percent of the time flipping a head in a tail would occur 25 percent of the time tail and then head 25 percent tail tail will be 25 percent and now I can change my outcome my outcome is something as a researcher that I state I want to know what the probability is of at least one head that becomes my outcome now so you see this idea of an outcome as we fluid that is my research question and in those cases well three of those have at least one head in it the first one has two heads in it that is at least one head the second one is head tails and the second one tail heads each of them has a head in it so only the last one tail tail that has no heads in it so if my outcome is the probability of at least one head that is a probability of 75 percent the 75 percent probability of at least one head and you can see starting to develop the notation here p and then in parentheses at least one head later on we won't use a full word we'll just use a symbol to to mean at least one head so the probability of at least one head is 75 okay we researchers we can change our outcome this time I'm going to flip my coin three times and my experiment is flipping a coin my event is recording you know what have what came in my outcome is I want all three here so I've got all these different outcomes head head head head tail head tail head tail head head tail tail tail tail tail tail tail tail tail so now my question can also be what is the probability of at least one head that's now my outcome of interest of all my outcomes this is my research question I want to know what the probability of at least one head is and if we quickly look down there we see it's again it's only this last one so the probability will be seven out of the eight so if we say seven divided by eight of course we're going to get 0.875 so 87.5 percent that is my theoretical probability so let's do an experiment so we're going to set up it's going to run some code here let's have a look at that let's see numpy.random.choice and what are we going to choose from from this array this numpy array with zero and one so let's call that tail and head and I want three of those and I say replace equals two and we've got to do that because there's only two choices there and I can't select zero the first time one the second time then there's nothing left if the coin is flipped and it's a zero we throw that zero back into the bag to be randomly selected the next time so let's do that if I were to do that we're going to see zero one zero so that would be tail head tail and if you run it again you're going to get something different so let's do this five times and I'm going to do this with list comprehension so list remember that goes inside of a set of square brackets I'm going to say numpy.random.choice in this array zero to one three times replace equals two for i in range five so that's going to be zero one two three four so it's going to happen five times so let's see at random what our coin flips were it was tail tail head head head tail head tail tail head tail head head that's what happened okay fair enough we understand how the code works let's do this many more times but what we want to keep track of is how many times there's at least one head now what we can do of course we can just sum over that each of those little arrays and if it's one or more you know or at least greater than zero that signifies there was at least one head in there so that's what we're going to do so I'm going to see the set of random number generator with the integer two that's the choice I'm going to have a counter count and I'm seeing that two zero and then I'm going to say for i in range one thousand so we're going to run through this for loop a thousand times I'm going to say flip underscore three is the sum of that little random values that I have and then if the flip is this flip underscore three is larger than one I'm going to increase my counter by one so that I know that at least there was one head in that specific set of three numbers and then I'm just printing the results out to the screen and so let's run that and see how close we get to this idea of 87.5 percent and there we go we see a total number of experiments was a thousand and the total number of at least one heads that's my counter that kept on counting every time the value was larger than zero so there was at least one head in there it was 87 877 so that gives me a probability of 0.877 so my empirical my results here my actual values for my experiment is very close to the theoretical 87.5 percent and that's from a thousand times rolling the dice three times so you start getting this idea of the theory what it should look like and then our experiment is not quite there because it really depends on the sample size how many times we do this now one little thing I think you must always be this is the something you just absolutely have to know probabilities range from zero to one you can't have a probability of something happening minus three percent of the time that this that that's not how it works and you can't have the probability of something happening 110 percent of the times it's from zero percent to 100 percent inclusive or in fraction terms a zero to one zero point zero to one point zero and if I have all these events and I you know put them together it must add to one as well so the probability of something occurring an event occurring plus the probability of the event event not occurring must also be one so the probability of the event not occurring is the one minus the probability of the event occurring such a simple thing then equation one but those two facts and you know the probabilities must be between zero and one and they must sum to one and so the probability of something not happening must be the one minus the probability of it happening because that brings us to this idea of negation which I want just to use equation one then in a little example so we're going to roll our fair die again and we're going to ask what is the probability of rolling a six given a single roll of course it's one out of six but then what is the probability of at least one six in two consecutive rolls so only one of them need to be a six they didn't both need to be a six then what about the probability of there being at least one six if i roll it three times or four times or more times well what we're going to use here is this this idea of equation number one there and we use this idea of the event not occurring instead of the event occurring sometimes it's useful to look at the event not occurring so what is the probability of not a six so that's symbol you see there in equation two that's a mathematical notation for not a six the probability of not a six well that's going to be one minus the probability of a six the event not occurring is one minus the event occurring so one minus one point one over six that's five five six so that's the probability of not rolling a six and what we can do here is to start squaring this if i roll it twice it's one minus the probability of not a six squared so one minus five over six squared so that's eleven out of 36 and the probability of a six in n rolls then is one minus five over six squared so let's just look at the code there so i'm going to say computer variable six one minus five over six to the power n for n and range one to eleven so one to ten times so the probability of at least one six in a single roll is one over six it goes up to 30 percent if you do it twice and if you do it ten times in a row the probability that there will be a six at least one six in those ten rolls are actually 83.8 percent so we can do a little scatter plot of that let's just have a look at that and you can see the more rolls i have the higher the likelihood that's going to be at least one six by the time we get to 40 rolls it's 99.9 percent that there will at least be one six in there so just remember sometimes we have to use negation okay so that's all fun you know you can play around with that play around with some code see what you can come up with more importantly we've got to talk about types of probability and because we're going to work with this kind of thing and the first one we're going to do is is unconditional probability so we're actually going to talk about unconditional and then also we're going to talk about joint probability and then if we scroll down a little bit and there we go our conditional probability so let's talk about unconditional probability i'm going to create a little scenario here we have two containers and in container number one there is some red balls some green balls and blue balls and the same goes for container number two and we're going to look at the relative frequency and i'm just going to set up a little data frame here with hot coded values so let's have a look at this on the left hand side i have container number one and container number two and this is the relative frequency so in container one red balls that makes up 26 percent of the total of all both containers in both balls so it's relative frequency 36 percent green balls container one 18 percent it's a blue ball in container one nine percent it's a red ball in container two seven percent it's a green ball in container two and only four percent it's a blue ball in container two so if i sum all of these up these are relative frequencies so relative probabilities there that has to sum to one and that's exactly those six values sum to one so no problem there but now i can also look at the sum totals for the different rows so let's do df dot sum and i'm going to set axis equals one so what's that that's going to do for me it's going to just add up all you know the column values for each of the two rows that's what axis one means so 80 percent of the balls are in container one and only 20 percent of the balls are in container two but if i do it by axis equals zero that's going to be the other way around so now we're going to get the this relative frequency of the balls 35 percent of them are red 43 are green and 22 are blue so down the margins we can write container one is 80 percent container two is 20 percent and if we go down the bottom we can say red is going to be 35 percent green is going to be 43 percent and blue is going to be 22 percent and we can just sum up these totals because they relative frequencies of the whole so 0.26 and 0.09 is going to be 0.35 and 0.18 and 0.04 that's 0.22 so the blue balls would be 0.22 or we can sum along these so this is quite common to have this sort of thing so we you know we've got to spend some time on that so now we're going to talk about this unconditional probability we see the unconditional probability of a green ball is 0.43 because it doesn't matter which container it is in if i just look at that it's it's a green ball it's the 43 percent chance because the 35 percent chance it's a red ball and 22 percent chance it's a blue ball if those balls were all put together it's independent of the container that it's in now we're going to talk about the joint probability though so we are interested in two outcomes here two things have to be they are joined it's got to be a certain color ball from a certain from a certain container so let's have a look at this if it's container number one and it is green it's the intersection of this idea of it being green and being in one and that's just our relative frequency there if we go to the chart it's green and it's in container one that's 0.36 that's the joint probability so both the container and the color joint probability both have got to be true is it on container one yes is it green yes they both have to be true that's 36 percent and that is our joint probability and now comes the important but conditional probability now that's the probability of an outcome given that something else has occurred in this time that other thing that has occurred has an influence on this thing occurring it's not like flipping two coins in the when you flip the the second time what what happened before is of no consequence now there is some consequence so this makes it infinitely more interesting to us so the question that we're going to ask here what is the probability that it is from container one if it's a green ball so you've got the two containers and someone you know your eyes are closed someone swaps them around around and you know randomly you take one ball out of it and i'm asking what is the problem and you open your eyes and it's a green ball so you're asking yourself what was the probability that this came from container one versus it came from container two remember if we ask it comes from container one that would be one minus it coming from container two but that's not what we're interested in here we're interested in what is the probability that it came from container one given that it was green of course that's going to matter because the relative frequencies are all over the show and in equation six we see this idea of conditional probability and so we say the probability of a given that b occurred that's equal to b and a having occurred divided by the probability of b so let's just have a look at the numerator there the probability of a and b a and b joint probability so this is the ratio of the joint probability divided by this unconditional probability at the bottom so it's the probability that it's in one and green we know that's 0.36 divided by the probability that it's green and the probability that it was green depends on nothing other there's blue balls red balls green balls and that was 43 percent so in the end that's going to be 0.36 divided by 0.43 as simple as that okay so a we're going to say it's it is container one and b that it's green so what is the probability of a it's from container one a it being green given that it's green you look at it in this green b so that's the joint probability a and b divided by the second one the given something has happened the probability of it being green so that's 0.36 divided by 0.43 so if we run that code you see the result there it's going to be 83.7 percent the probability that that comes from container one now container one at 80 percent of the balls but you know green was a certain likelihood as well so given that it is green it has suddenly raised the probability a little bit that this was actually from container number one remember then the probability of being from container two would be one minus 83.7 you know 100 percent minus 83.7 percent so how do we know if something was a conditional probability or if the two events were independent of each other we can actually test for that and what it would do is the probability of a given b we've just seen the equation for that here an equation six how to do that if that equals the probability of a and if it was probably of b given a that that is equal to the probability of b you know those two things are equal to each other then it was actually the events were independent of each other it did not matter what happened before what we were given so we see an equation two there that you know writing it out twice a given b is a and b over b and b given a is a and b over a so we've done this here let's have a look at it the probability of a given b a given b so that's that's this joint probability divided by the probability of b being 80 so probability of being in container one given that the ball is green and the probability of b given a so the probability of green ball given that it was from container one so this time you know what container came from what is the probability of it being green and then the probability of a green ball probability of b being from container one so we're just saving all of those up as as values and I'm asking is the probability of a given b equal to the probability of a no it's not is the probability of b given a equal to the probability of b no it's not so I can say these two events are dependent upon each other so that's how you would measure if if events were independent of each other so joint probability remembers means and so both have got to be true it's got to be green and it must be from container one but what if it's all so here we're talking more about the union of something it's in if you think of the union of two sets that's all it's either in one set or it's in the other set or it's in that intersection between the two that's what union means it can be in this combination of them it can be in one or the other one so what we're talking about here is the probability of it being an a or an b and think about a Venn diagram those two circles in the intersect somewhere now that union of those two is the probability that it's in the one and the probability that's in the other one or plus I mustn't say and that that might be confusing probability of one or the probability of the other one but remember what we said before that intersection those things get countered twice so we better subtract that intersection the probability of it being an a and b so the probability of a or b that's a probably of a probability of b sum those two up minus the intersection the joint probability so the probability of it being red we sum up over all the reds the probability that's two sum up over all the twos the probability of red given two is 0.09 and if we just sum all of those up and subtract the intersection we get 46 so that's the probability and this time I didn't say the colors I was interested in this time it's the red one or it's in container two so you draw it what is the probability that it's red or it comes from container number two so that's when we're going to use this idea of this Venn diagram and this union of them minus the intersection of them and then the last one or second last one is just what is the probability that our random ball is both red and in container two you can see the equations there but you can see it's only it's just simple algebra of what we have before because if I divide this first little one if I divided that by probability of b then I have again probability of a given b is the probability of a and b divided by the probability of b so it's just a very simple algebra from what we had before and if you run through this code what you're going to see well that's just the relative frequency isn't it it must be both red and in container two well if we go back to our little table there that's exactly what it was that that joint probability that's nothing other than the relative frequency and again we have this idea of the complement so if it doesn't occur it's one minus the probability of it occurring given some universal set so let's work through a very interesting example here and I wonder if you're going to get this one right um um if we were to play some bets I wonder what uh not that I'm a betting person at all or condone it in any way but fun nonetheless to think about what you're going to select as a solution so imagine we have three cards one is red on both sides one is white on one side uh a white on both sides and the other one's red on one side and it is white on the other side so red red white white and red white now we're going to just shuffle him under the table and we're going to select a card that's chosen at random and you get to see one side and that one side is red and I ask you what's the probability that the other side is red so don't read any further just think about it don't read so red red white white red white and I pick one and I show you the one side you don't know what the other side is and at front side that you see is red what is the chance that the other one is red well let's just think it through with what we know about probabilities now so what are all the different outcomes that we could have from these cards it could be red red so I show you red and the other side is red it can show you red red again because there are two red sides think about it I can draw the card and show you red side number one for that red red card or I could have drawn it and shown you the other side of the red red card so that's two red reds think about it the red white card I could have picked it up and shown you the red side front and then the white would have been at the back or I could have picked it up and shown you the white side and the other side was red and then the same argument for the white white card I could have picked it up shown you the one side white and the other side was white or I could have picked it up and shown you the other white side and the other side was white. So there's actually six outcomes. So now I think about it. If I show you a red, that's in the front, that's going to be one of those first three ones. And look at what it's at the back. There's a red at the back, red at the back, or white at the back. So the probability that the other side was also red is two-thirds, and I think many people would have said it's a half, but it's not. It's two-thirds. So that's just a little teaser there to mess with your mind, and congratulations if you've got two-thirds the first time round. So always come up with all the possible outcomes. That's what it's all about. Come up with all the possible outcomes. So now I want to talk to you about random variables. So a random variable is a function, a mathematical function, and that maps the outcome of an experiment to a number. So we're just actually in our spreadsheet capturing a number. And by number I'm using sort of a loose term. I can encode a categorical variable as a number. You know, so it doesn't matter what the variable is. Let's just call anything that I then capture. Let's just call that a number. So very abstract definition here. So I'm just mapping the outcome of my experiment to a number.  So there are all the outcomes if you think about it. I can, you know, both can be one and that's a two. I can have a one and a two that's a three, or it can be a two and a one and that's three. I can roll a two and a two, that's four, a one and a three and that's four, and a three and a one that's four. And for a five I can roll a two and a three, and a three and a two, and a one and a four, and a four and a one. So you can look, have a look at that. There's only one way to get a two. There's no ways to get a one. I mean that one dice, you're rolling two dice. So you can't sum two one. But there's only one way to get two. And right on the other side of the spectrum, there's only one way to get a 12, a six and a six. But if you look at a three, there are two ways to get a three with these two dice. One can be a one, and the other one a two, or the first one can be a two, and the second one can be a one.  Two and two, one and three and three and one. And there's one, two, three, four ways to get a five. And there's one, two, three, four, five ways to get a six. And there's six ways to get a seven. And then it drops down again. Fewer ways to get eight, fewer ways to get nine, fewer to get 10, two to get 11, one to get 12. So it's most common to get a seven. So there's this pattern to this random variable of us. So I write it down. So what we would say our random variable is x. So it gets a bit confusing. That x is just going to be our column header. That becomes our variable. And the values that we jot down, those are random variables. So it gets slightly confusing. So I try to stick to the fact that we call that column header in our spreadsheet. That's our variable name. And we have data point values for that. But each one of those values technically, that is a random variable because it maps an outcome to a number. So the probability that x is two, that our random variable, x equals two, that's one over 36, because there's 36 of these, one, two, three, four, five, six, 36 possible outcomes. So two, there's only one way to get a two. For three, there's two ways to get a three. And up to the seven, there's six ways to get a seven. And then it drops down to one way to get a 12 all the way in the end. So you can just run that little list comprehension there. And we can see the probability. So the highest probability was a seven at 16.67%. So let's just plot that as a little bar plot because remember these are discrete variables. And you can see this nice, nice distribution. So for a seven there, that was the highest probability of occurring that occurred most of the time. And now we can start thinking about probability in a new way. We can think about it graphically, because I can ask, my outcome could be what was the probability of my random variable being 10 or more? So this is how we're going to write it, probability my random variable x being greater than or equal to 10. Well, I just sum up of these, the one for 10, 0.0833 plus 0.0556 plus 0.0277. I can add the area of these rectangles. I want you to start thinking about that. So I can say 0.083, 0.055, 0.028. So I'm just rounding off there a little bit. So I can now start saying that the probability of my outcome, my outcome of interest is a 10 or more. If I roll to die, what is the probability that the score is going to be 10 or more for the summing up those two face values? Oh, that's 16.6%. 16.6%. So just start thinking about that. So I want to start thinking about this idea again of a theoretical distribution and an empirical distribution. So Sydney, this was theoretical. We worked this out by pure numbers. So let's roll some actual die or at least we're going to simulate them. And this is going to give us an empirical distribution versus our theoretical distribution that we saw there. So I'm going to set the random seed, the pseudo random number generator. I'm going to see that with an integer of 3. I'm going to say roll underscore totals and I'm going to create a data frame object. And I'm going to have one column. So I'm setting this up as a Python dictionary with a key value pair. My key is going to be the column header. That's roll total. And I'm going to do list comprehension 10,000 times. I'm going to draw from this random dot rand int. And rand int is almost like choice. I'm just giving it the lowest value and the highest value low of one high of seven. Remember, this is Python. The one is not going to be included. So it's actually going to be one, two, three, four, five, six that it chooses from. So that's just a different way of doing this random choice thing, using rand int, random integer between a whole number. In other words, between one and seven, but seven's not included. So it's one, two, three, four, five, six. Choose two of them. And then I'm passing that to the sum function. So I'm summing those two. So I'm just simulating a rolling two dice. And I want that to be converted to a list. And I do that for I in range 10,000. So I'm doing that 10,000 times. And let's look at those sum totals then. So the first time I rolled a four, then a six, then a two, then a seven, then a 10. So you get the idea. So let's do value counts to see, you know, how close we got to this. And lo and behold, seven was most likely at almost 17%. So, you know, not too bad. But let's just sort it by this index rather than the mode, the most common first. So I'm going to use the dot sort index method for this. And that's going to sort by the index. So it's going to go two, three, four, five to 12. So you can see this idea, 1,699 of my 10,000 rolls. That was a seven. And it was very unlikely to roll a two or 12. So let's just create a bar chart of that. So this is going to be my empirical distribution. And it looks very much like my theoretical distribution. So there's my empirical distribution of values. And again, I can ask what was the probability of there being a 10 or 11 or 12. So I'm just going to sum up. There was 854 of those, 559 of those, 299 of those. Sum those up divided by 10,000. That's the probability of a 10 or more in my empirical experiment that I set up here. Good. So let's talk about distributions then. It's this pattern that we get from our data, how many times occurred, their frequency or then their relative frequency. So imagine now I'm going to do the following thing. I'm going to set up this normal distribution. So without us me explaining explicitly what a normal distribution is, I think you've got quite a fair understanding of it already. Most people have that intuitively, this bell shaped curve. So near the middle, near the mean or the average, it's more likely to have values there. And as they go away with a symmetric bell shaped curve towards the smaller side and the bigger side, those values get less. So there's the idea of a mean and a standard deviation. So we have this norm dot r vs function in the stats module of sci-pi. So stats dot norm dot r vs. It's going to work very much like the numpy dot random. It's actually just slightly more powerful. There's a few more things you can do, but it's really nothing other than numpy dot random dot one of the random functions that you do get. But here you get stats dot norm dot r vs. And I can set the loc location. That's a keyword for mean. So I'm setting a mean of 160 scale is the keyword for standard deviation. So standard deviation of 10. I want 200 values, please. And I'm sitting a random underscore state. That's the same as sitting numpy dot random dot seed. Yeah, we have an argument in the r vs function, a random seed. And I'm going to assign that to the computer variable height. So what I'm imagining here is 200 people and the mean of their heights is 160. The standard deviation is 10. Just from that distribution, please draw me 200 random values. And there we have it. So let's draw histogram. Now height is now continuous numerical variable. So we're not interested. We're not interested in a bar chart anymore. We now want histogram. So there's our histogram. And you can see I'll plotly decided that it's going to do these steps of five centimeter increases 150 to 154.99. And the count of this was made up 0.135 of all the values. So let's go to the top there between 160 and 165 or then 164.999. It made up 22% of the total. So this is a nice little histogram. So the reason why that happens is because I set his norm equals probability. So we don't get the actual values. We get this relative frequency. So I just want to show you this little other plot. I didn't think it's going to be around that much longer. This figure factory, they are changing plotly changing it. But we get create the create underscore this plot that's going to do exactly the same thing we're going to hit the histogram. But I've also done this nice little curve. It's kernel density estimate. So it gives us this nice little bell shaped curve instead of this blockiness, of course, of our histogram. But it's trying to tell us the same thing other than the fact. Remember that this is an artificial binning of our values. They're actually a continuous numerical variable. So in the end, this is really what we want to deal with is this smooth curve. And we're going to see a lot of these smooth curves. And you're going to understand exactly what that means. What we're trying to get to here is this, what I said right in the beginning, there's this empirical distribution and the theoretical distribution. And there's this law of averages and this law of averages that really states that our theoretical or empirical distribution that is going to start approximating our theoretical distribution. So empirical distribution is going to start to approximate or get close to the theoretical distribution if our sample size gets bigger and bigger and bigger. And that's something I think most of us in research also, you know, have heard a million times and we sort of understand that the larger sample size, the closer we are going to get to what the actual values are. So this idea of the law of averages. So let's look at this at play a little bit. So what we're going to do here is something very interesting here at the end. I'm going to simulate the population of 10,000 subjects. So I'm going to imagine in a lab or working with some economic data or some human data, it doesn't matter what it is, some astronomical data really doesn't matter. I have 10,000 subjects and I'm going to measure this random variable in them. So this is going to be this variable and we know what it is in each and every one. Just imagine we know what the value is in each and every one of those 10,000 subjects. And we're going to make as if it comes from what we call a chi-square distribution with degrees of freedom of three. Don't worry about that at all. So let's just set that up and that's my population. So in my population, there's a certain variable and we measure them and we know what it is in all 10,000. Hardly ever do we know this in real life, but this is a computer. We can simulate these things. So random dot C dot 42. And then I'm going to use the round function and just round to one decimal place just to make things easy. But I'm using numpy dot random dot chi-square and there's its parameter three, three degrees of freedom and I want 10,000 of those. So I said, don't worry what it means. That's not important, but I want to show you what it looks like. So here's a histogram of this variable for all 10,000. You can see what a chi-square distribution is. It goes up till here about maximum here and then sort of has a long tail on the right hand side. So we know this, we know what this is for the actual whole population. But now as researchers, we can't get 10,000 subjects in our study. We only select at random 30 of those individuals. So 30 subjects from our whole population of stars or whatever it is. We only take 30 of them and let's look at a histogram of our 30. And sort of we can imagine it forms a chi-square distribution to some extent. But from a sample of 30, you know, we're not quite sure. But what we're going to do now is we're going to increase that sample size. And of that population, I'm increasing the size of my study and I select 100 samples. So let's do that, 100 samples. And now it looks almost like some other distributions, but sort of suppose we can start seeing chi-square. So let's ramp it up to 10,000. And now we see the chi-square distribution starting to come out. So my empirical distribution is starting to resemble my theoretical distribution. Now, it wasn't really theoretical because my whole population was also actually only a sample from something that was theoretical. But I think you get the point, the larger the sample size, the more it's going to approximate the population from which those values were taken. And now for something really interesting. And this is really going to be one of the most important topics in data science. We're going to talk about sampling distributions. So let's stick to our population that we had. And we're going to take 30 samples from our whole population of 10,000 subjects. And what we're going to do is we're going to calculate the mean of our tiny little samples. Have a look at this. We went out, we planned our research, we took it to an ethics committee. If that's what you have to do or no, we submitted it some way for approval and we got a grant to do it. And we're going off now and all we could do because it's very expensive, either in time or human resources or finances, we can only take 30 from that. And we take our 30 samples from this population of 10,000. We have this idea of mapping our event to some value that we can do our random variables. In other words, we collect this value for this variable and this 30 sample size of ours. And we get a mean of 3.183. 3.183. And that's what we're going to report on. That's what we have. But, you know, is this representative of the actual population? Well, presumably we took this at random. There was no bias in how we selected that. But, you know, we're not quite sure and we need to, you know, we need to do some statistical analysis, perhaps on this, you know, express how certain we are that 3.18 is actually the mean of our population. Now, I want you to imagine you have infinite riches.  Every week, someone gave you lots of time, lots of resources, lots of money and you do an experiment again. And the next week again. And the next week again. And every time you take 50 or 30 sample of 30, you know, at random, some of them might be the same one. Some might be different ones. But you're doing this 50 times over. Every time selecting 30 and every time you select your 30, you calculate the mean of your 30.  You're actually going to have a distribution of means. And that's what we call a sampling distribution. The sampling distribution is this pattern of a statistic. Remember, a statistic is something we calculate. When we did a summary statistics or descriptive statistics, a statistic is a value from a sample. A value from a population. So the mean of a population, that's a parameter. But some calculation point estimate or measure of dispersion from a sample that's called a statistic. So if you had a bunch of statistics that they form a distribution, and that's exactly what we're going to imagine we have the power to do here. So I'm going to create this computer variable mean underscore 50. I'm going to use this comprehension there. Let's see what it is. It's going to calculate the mean of a random choice of 30 from my population. And it's going to do that 50 times. So before I end range 50. So what I'm going to have now is I can imagine, as I said, all the riches in the world and I can do my experiment, my research 50 times over. And let's see. Remember when I did it once, my mean was 3.183. Now every time it's going to be slightly different because I'm going to have different members from the population in this. So let's have a look at what the histogram looks like now. So if I do this 50 times over. Now that looks a bit strange because this distribution does not look like a chi-square distribution anymore. The empirical distribution of the actual value. Those are empirical distributions of the actual value. This is a sampling distribution of a statistic, not of the actual values. So let's ramp that up. And now I'm really rich and I've got oodles of time and resources. I'm going to do this a thousand times over. So I'm going to imagine that I can do my exact same research a thousand times over every time I choose 30. And have a look at what happens here. This is starting to look like a normal distribution. A sampling distribution of the means. So the sampling distribution of a statistic and you can do it for standard deviation, for variance, whatever you like. There's going to be a pattern to that as well. And that eventually is going to allow us to do some data science to understand the story that's hidden in our data. Our result, because we can only do our research once, is going to be one of many possible outcomes. And that is going to tell us if our outcome that we found was a rare event or it was a very common event. And that is what we're building towards.


One of the first steps that you need to take in the process of creating consistency is to start noticing what you're doing. Start observing your own thoughts, your own words, your own actions. Every morning, when I'm getting ready to trade before the market opens, I'm actually writing down, like not typing on a computer, physically writing down how I'm feeling. You know, was there, did I have an argument with one of my kids or my spouse before I started trading? Am I feeling tired because I didn't get enough sleep the night before? Have I drank too much caffeine that morning? Do I have some type of project or deadline or something that I'm working on that's going to be taking away or distracting me from my trading? So I write these things down every morning just to get that self-awareness flowing so that I start to think and observe myself and how I'm acting and how I'm thinking. Because remember, everything that we do or think or say as a trader, it contributes to, you know, our belief, our mental system. And what this will do is as you continue to get really good at self-awareness and observing your own thoughts and words and actions, this is going to help you avoid errors because all of a sudden you're going to start catching yourself thinking a certain way. You're going to start catching yourself acting a certain way. And when I say acting, I mean pressing that confirm and send button to send trades, you're going to start saying, wow, should I really do that? I'm thinking in a way that does not correspond with my goals and my desires. Should I really do this? And when you start catching yourself, you're going to be able to start avoiding some of these errors that you were previously taking. And lastly, so you don't create this kind of emotional pain that some of us do to ourselves in our own minds is the last piece is don't judge yourself. Understand that we are still going to make mistakes. Everyone makes mistakes. We are human. And so you can't judge yourself too harshly. So it's a balance of observing yourself, catching yourself in the act, and then correcting yourself without judging yourself too harshly.


In Kenya's Darbid refugee camp, the thousands of new Somali refugees are struggling as food insecurity worsens. As Somali suffers its war drought in years, Somali refugees have fled over the border to Kenya. In recent months, 135,000 have arrived at Darbid, one of the world's largest refugee camps located 90 kilometers from the Somali border. The camp is home to 360,000 registered refugees are many more unregistered. In February, the Kenyan government resumed refugee registrations at the camp, allowing the new arrivals access to food aid. The World Food Programme says rations have been caught from 80 percent to 60 percent of the daily nutritional requirement due to reduced donor funding. The immediate impacts we see is household food insecurity because families that used to prepare probably three meals a day have now reduced to prepare either two meals or a meal a day. And that's quite extreme. We also observe families from our monitoring who are now increasingly borrowing food from friends, relatives, or even begging for food because the food that we give is not able to last the 30 days, but it takes about 15 to less than 20 days. We are facing difficulties with the funding landscape. We have fewer resources to be able to manage these cases, including human resource, but also the commodities that we use to treat malnutrition. How we're handling it right now is, well, we have to work with the little that we have. And as the International Mercy Committee, we're trying to come up with innovative solutions to handle this influx and still be able to give our clients the care that they need and help them to recover from these challenges that they have with their nutritional status.


So I sold a total of 2,997 tickets in 2021. That's a pretty good number, but that turns into sale of... What's up, guys? Welcome back. My name is Trez. Here on Thumbs Up, we're going to be talking about buying tickets, selling tickets, making sure that you have all fun with your tickets. We're going to be talking about my 2021 year in terms of how many tickets I actually sold and how much money I actually made. I used to just leave little tidbits here and there in different videos talking about specific events, how they did, specific teams, maybe how I've done with them in the past. But I've never talked about how much I've actually made in a full year, but I've tallied it all up for you guys, so I figured this would be the time to do so. Really, I'm just trying to be transparent about everything I do. It's not because I want to brag about how well I've done or anything like that. It's really more just to kind of give you insight into how the economics actually work for a small-time ticket buyer, and then also to kind of inspire you to see, you know, is this something that you might actually want to consider doing yourself? I do want to make it clear, though, that, again, this has taken me many, many years to get to this point. I started way back in 2013 selling my first pair of tickets, and then it's kind of grown since that point all the way until now. So it's been over seven years of me growing my ticket base year after year, so that's kind of where we're getting to at this point. There's definitely a lot of hard work involved. There's definitely some luck involved. But if you put in the time and effort, there's definitely a possibility to go ahead and make similar-style numbers. Though that being said, though, I do want to lay out a couple of ground rules. One, I'm going to be listing everything in USD. I personally am in Canada, so I will be reporting on my taxes in Canadian dollars, but for the purpose of this video, a lot of my viewers are in the U.S., so it kind of makes it a bit easier to make it, you know, understandable across the board. And number two, I'm going to break it out monthly. So each month, you're going to see how much I made in terms of ticket sales, how much those tickets cost, and how much I actually spent on buying new tickets that month. Enough of the rambling. Let's talk numbers. So in January, it was an amazing month. I sold a total of two tickets. I sold two tickets, total revenue on that $2,412. Now, there is one thing to mention. I actually spent a total of $2,200 on those tickets, but I made a little error. I made a mistake. What happened was I sold some tickets to the College Football National Championship and I had a pack of four seats, but I sold them as a pair of two and two. Unfortunately, I couldn't sell them that way because there were social distancing rules in place. So had I sold those two seats together and then sold the other two seats together, I would have run into a big problem where one or both of the buyers could have come back saying, hey, these seats were not socially distanced. I want my money back. And they would be completely in the right. So what I ended up doing, I ended up selling the pair of seats. I ended up not selling the other two tickets, so I just ate that cost. But in the end, we still made a total of $212 net. So not too shabby. In the month as well, I bought a total of four tickets and they were those same ones. Spent a total of $2,200. Now February was even more exciting than January. I sold a total of zero tickets. Bought zero tickets, sold zero tickets, nothing happened. There is no activity whatsoever. I will mention the Super Bowl did occur in February, but I unfortunately do not own any Super Bowl seats. But I unfortunately had to stay on the sidelines. Now March is where things start to get a little more exciting. So let's start off on the ticket purchase site. How much did I actually buy in the month? So in March, I bought a total of 76 tickets where I paid $1,804. That's how much it cost me to buy all those seats. Most of them were MLB related. MLB at this point had decided that they would be opening up their venues back to getting some people in the stands and it would be socially distanced and limited seat capacity. So quite a few restrictions in place for those who could attend. But for the ones who could attend, they were able to actually start going to go and see some sports. So people were excited to say the least. At this time, I didn't quite know what to expect in terms of who was going to be going, how many people would be going, what kind of prices things would be selling for. So I was a little hesitant at the start. And even Toronto Blue Jays, they still weren't going to be playing at home. They were still playing out in Dunedin. So there was quite a bit of fear in the marketplace. So I committed a little bit in terms of buying some MLB seats, but I didn't go too hard. However, that was a mistake on my part. I sold them a total of 52 tickets in March for a grand total of $7,003. That's how much I made in sales. Ticket cost on those seats $1,236. Net of $5,767 in that month alone. Yes, you get the bump for home openers and baseball every year, but the amount of demand that there was for these seats, it was just mind-boggling off the charts. I've never seen anything like it before and likely will never see anything like that ever again. So had I committed even more, that number could have been much higher. But just seeing that kind of gave me an idea of like, okay, maybe people are really ready to get back in action. And I knew that next month I wasn't going to make the same mistake again. So now we're getting to Q2, April. April, we sold 118 tickets for a grand total of $10,318 in ticket sales. Those tickets came to a cost of $3,808, netting $6,510 for that month. Bulkers, absolutely bulkers. So in that month as well I bought another 230 tickets for a total of $14,106. So I was committing to buying more and more tickets. Of those 230 tickets though, I only bought one pair to the biggest actor that month, Bad Bunny. There was some crazy demand for once tickets actually started going on sale and they hit the resale market. Those $40, $50, $60 seats were going for anywhere from $150 to $350 or more and even higher as you got onto the floor. It was madness. I didn't know who he was. I don't know any about his type of music. So it was a completely new learning experience for me and I completely, completely, completely missed out on a huge opportunity. But overall month still went very, very well. Moving on into May, we sold another 203 tickets for a total of $38,457. Those tickets came at a cost of $20,036, netting $18,421 for the month. In that month again, ticket purchases were up. I bought another 548 seats for a total of $40,842 spent on new inventory. Now May had a lot of exciting things going on. Garth Brooks went back on sale, Billy Eilish went back on sale NFL schedule was released that month. Toronto Blue Jays, they were going up to Salem Field in Buffalo. So those tickets were going on sale. That was another monster event when those were announced. All crazy things were going on this month. So I was just buying, buying, buying, buying, buying, non-stop because there was just too much action to catch on to. The other thing I started noticing more in May was the seating of its vaccinated and non-vaccinated seating. So vaccinated seating, they had no social distancing whatsoever and the non-vaccinated seating, those were still socially distanced. As the months were going on, you kind of gained the idea that hey, maybe they're going to be starting to enforce more of these vaccination rules in the future, but at this time it was still, you know, you had the option because again, getting vaccines at that point was still a little difficult. But with all that was going on in May, there was still one event that was by far 100% the best event of the entire month and maybe even the entire year. NHL Playoffs. Toronto Maple Leafs. Montreal Canadiens. They got matched up first round. They hadn't played each other in many, many years and this game, game six, Montreal decided, hey, we're going to allow 2,500 people into the stands to go and see that game. Up to this point, no NHL games have been played in Canada at all at this point, so this was going to be huge. We're talking about playoffs, we're talking about, you know, moving on to the second round implications. We're talking about two Canadian rivalries, madness. And back in December 2020, I ended up just getting through the waiting list so I was able to actually get some seeds for round one for the Montreal Canadiens. How much did I make for them? Got away to the end to see that total. But let me tell you, it was huge. Moving on into June, I sold another 110 tickets for a total of $13,847. Those tickets cost me $7,284 netting $6,562 for the month. Also in that month, I bought another 116 tickets. Total cost $12,862. This was another awesome month. Bruce Springsteen announced a whole bunch of new dates I was able to capitalize on. Montreal Canadian Playoff Tickets, that was still in run because they somehow managed to win in the first round and the second round and the third round. Wildness. Honestly, completely unexpected. Then we continue on to Toronto Blue Jays and Buffalo continued their amazing success in terms of ticket sales. Everything together is just insane. And also though, another change that we started seeing MLB teams were moving to adding more capacity, higher attendance. Slowly we all started seeing the move towards more vaccinated seats, so a lot of changes were taking place. End of Q2, we're moving into Q3. Started in July. In July, I sold a total of 200 tickets for a total of $20,319. Those tickets cost me $13,338 for a net of $6,981 for that month. July also spent quite a bit, a bunch of new tickets, $334 to be exact, for $24,362. Both the NHL and NBA were having their finals this month so those both wrapped up. I was able to get some Stanley Cup finals tickets. I was able to get some NBA finals tickets. The Rolling Stones also announced a bunch of tours that month. Metallica announced a bunch of new dates and the Toronto Blue Jays were finally coming back to Toronto. Home opener was there. You can go check out the video where I actually went back to that first home opener, first game in almost 600 bajillion days or whatever it was. It was not quite what I was expecting. They only had 15,000 people in the stands, but it was still still one. They won, which is always awesome. And two, it was the return of something that we hadn't seen in basically two full years. So it looked like things were in moving in the right direction and we're going to start seeing more and more events getting announced, going on sale, and making a ton of money for the rest of that year. So in August, things were still rolling on. I sold another 319 tickets for a total of $30,754. Those tickets cost $18,442 for a net of $12,312 in this month. That's a lot, a lot of money, a lot, a lot of money. I bought another 509 tickets for a total of $30,292. In this month, I actually identified an event that might have potential for a three to five extra turn. So I decided to go hard at this bet and I spent over $16,000 just securing tickets for the entire run of that production. The rest of this month is just about selling tickets for the NFL, CFL, college football, MOB tickets, and then even seeing more and more new concerts going up on sale. At this point, all systems are a go. It's just a matter of trying to keep up with everything that's happening, seeing what's going on sale, buying those tickets, selling those tickets, and just trying to keep up with the demand. As we roll into September, another 208 tickets got sold for a total of $26,653. I bought another 289 tickets that month for $16,536. This month, I also had a bunch of major things going on sale. There's Tool, San Minha, Trevor Noah, Boo Fighters, it was just one after the other. Also, I was in Vancouver for a week and a half visiting some family, so I also lost a bit of time there, but I was out there still able to manage some Luke Holmes tickets for some of his Canadian dates. Also this month, I started my ticket discord group links down below. If you guys are interested, it is an opportunity for you to also buy along with me, see what I'm buying, see what I'm interested in, and getting some insight into what to buy next. Now it was a lot of work, but now we're getting into Q4, the big push for the end of the year. We're into October. October, we sold a total of 396 tickets for sales of $44,160. We're getting some really big numbers now. I bought another 668 tickets this month for a total of $45,337. This was another big month for Ontario because the government finally announced that they could open up indoor venues to 100% seating capacity. As I made NHL was back on, NBA was back on, all those great invets indoors were going to be taking place in Ontario.  Love me some MLB playoffs. Those again, great, great buys. And there's one other thing I noticed actually is that tickets in Ontario, the demand for them was not quite the same as it was in separate markets in the US of A. Toronto Maple Leafs historically don't have many season seats come up for sale, let alone many individual seats for the regular season. So it was a surprise to me when I got an email from my account manager saying, hey, we're selling half season seats for the Toronto Maple Leafs. Are you interested? And I'm thinking I saw, yes, yes I am interested.  So obviously I wouldn't bought them, but I just was a little taken aback thinking to myself, you know, this is not generally a normal thing because there's too many people who want these seats, there's always so much demand for them, so what's going on? So you could see even in the resale market for Toronto Maple Leafs tickets, yes, there was that huge premium, but that premium was not quite as large as it used to be in the past, and there was even more availability than normal. There are no complaints whatsoever. I'm going to enjoy those half season seats. Oh, also, also, have you heard of this BTS? Well, they also went on sale this month, a huge worldwide phenomenon as always, and I had 0.00 tickets. I actually personally didn't even bother trying because I just know how insanely crazy it is to buy BTS tickets. I don't want to be on the wrong side of those fans, so I didn't even touch it. You could see the prices that some of these seats were going for, you bought them for 80 bucks, you sold them for $700, it's wild, okay? Last thing that happened this month is when I bought those Bruno Mars tickets, but more on that in a little bit. Now, November, this was by far the biggest month I had the entire, entire year. Total sales and tickets, $779. Total dollars in tickets sold $82,781. Ticket cost $56,547. Net of $26,234, so it was a massive, massive month. I cannot understand how massive this month was. In terms of tickets purchased, I bought another $2,204 seats for a total of $102,302. This was the first month ever in my entire life where I spent over $100,000 buying new tickets. That is a lot of inventory. So, what happened this month? Well, first off, Morgan Wallen. Oh my goodness, this was wild. Morgan Wallen was, I don't even know how to explain it. So, you know how I saw a bad buddy back in March? Well, I wasn't gonna make that same mistake again, so I went all in on Morgan Wallen. Sale after sale after sale from Morgan Wallen, it was just non-stop. You could basically buy almost anything in the building and you were gonna see some massive, massive returns almost instantly. The rest of the year was a lot of MLB renewals and I was even able to purchase some Chicago Cubs season seats. I had been waiting almost 7 years for these seats, so it was a bit of a surprise when that call finally came in. But I heard the call and I bought. Lastly, December. I sold another 538 tickets for $62,617. Those tickets cost me $42,240 for a net of $20,377. Wow. That's a big wow. Also, ticket purchases 260 new tickets for a total of $29,291. Big one for December that I do want to mention though, Adele. Adele was probably by far the wildest event of this year. See, prices start somewhere like $200, like $1,500, and then you were seeing them resold from anywhere from $1,500 to $10,000 plus. It was bonkers, wild, crazy, just absolutely nuts. People were paying anything and everything for them. And, you know, I don't know what else to say.  So those were a lot of the things that kind of kept me busy buying this month. Unfortunately though, this was also the month where I went to Las Vegas for Bruno Mars and where I lost over $1,600. If you guys are curious about that video, go ahead, take a look in the description. I left a link there. You can kind of go see it all. Get sad with me. It's, it's a depressing story, but it happened. So what to do? All in all, though, this was a monumental year in many ways. One, the way it started off, it could have easily gone the way in 2020 where I actually lost money in the year. But in the end, it ended up actually being one of my best years ever. The way social distancing worked at the beginning, all the demand that was there for events going on, and then just the volume, the sheer volume of amount of events that went on sale to be sold, there was a lot of opportunity to make a ton of money this year. So I'm very grateful for how it all kind of turned out in that regard. Now, I do put an asterisk on this month specifically and even going back a couple of months with the new variant. It is causing cancellations and postponements across all these different leagues and concerts and venues. So it is becoming a problem in terms of whether those events will actually take place. I try my beliefs out of Senators, Toronto Raptors. They've already announced that a bunch of their events have either been postponed or cancelled. So a lot of sales that took place for those events are going to have to get refunded unfortunately because they're just not going to take place. So with the full year totals, let's summarize, shall we? So I sold a total of $2,997 tickets in 2021. That's a pretty good number. But that turns into sale of $339,321. That's a lot of money. That's very, very good year. I was really surprised to be honest. It turned out the way it did. In the year I bought $5,238 tickets for a total of $319,934 spent. So a couple of things I want to mention as well. One, this was an anomaly type of situation this year. I don't think it will ever be replicated ever again quite with these type of situations and circumstances so I was able to capitalize on what occurred in the year. So that's the first thing. So don't expect these ever to continue every single year. Number two, like I mentioned already, cancellations were not taken into account with the totals at hand. As events are canceled, some of these sales will have to get refunded. So some of these numbers may not actually be the final true total. So it might get reduced a little bit as well. Number three, the other main one is that I did not factor any of my expenses with purchasing these tickets. Yes, I have the ticket costs themselves. I also factored in the sales commissions that are associated with these sales. But it doesn't pick up any of the costs associated with fan clubs, with travel, subscription costs, processing fees, anything like that. There are quite a few expenses that I do have to operate this business. So none of that is factored in. Give this video a like or comment down below. Let me know if that's all you guys are interested in. And lastly, number four, I do need to pay taxes on this number. So again, this number will be brought down even further once all that is filed and paid. Now, another thing, what was the absolute best one-off event that you had in the entire season where you made the most amount of money? As you might have already guessed, it was the Toronto Maple Leafs versus the Montreal Canadiens Game 6 at home in Montreal. I had four tickets available. Total of those four tickets combined, net me a profit of $2,995. Think about that. Four tickets turned into a profit of basically $3,000. That's $750 per ticket on average. That is wow. Restricted events, great rivalry, haven't seen hockey in a year and a half. Everything was working for it, so in the end it ended up being a very, very profitable event for the entire year. On the flip side, worst event of the year, unfortunately as you may have already guessed, Bruno Mars. Need I say more? $1,600 loss. Yeah, that is. I'm not going to talk about anymore. You can go ahead and see that video and learn all about it. It's too depressing for me to talk about again. You guys can check it out for yourself. Biggest surprise of the year, I would say, was just the speed of which people started to flock back to events. When the MLB announced that they would be doing limited attendance and socially distanced seating, I didn't think too much of it at that time. So when things did end up going on sale again, I didn't know what to expect, honestly. And the amount of demand that was there to go back to see those games was, again, mind boggling. Absolutely blew me out of the water. Did not expect that whatsoever. Definitely the biggest surprise of the entire year. In terms of missed opportunities, I would say it was a lot of those verified fan seats. Obviously, you have the BTS and the Adele's and the Olivia Rodrigues and there's so many that go on all the time. Hard to catch every single one and get verified every single time. So if you are able to get through on those verified fan pre-sales, generally you'll do very, very, very well. But on the other hand, if you miss out, it's not the end of the world. There's always going to be another one down the road. But there's a lot in terms of missed opportunities if you don't get through some of those verified sales because you just can't get tickets any other way. Well, I mean, you can buy them on the resale market, but besides the point. Definitely have a comment down below if you guys saw some tickets this year as well. Let me know what your best events were, what your worst events were and something new that you learned. I'm always curious to hear what you guys have to say, so definitely let me know down below. So with that being said, be sure to hit like button down below. Also be sure to subscribe. Have an already great new content every single week. See you guys next time.


I wanted to ask you about the energy consumption in mining. What is the solution for it? Well, the energy consumption in mining is misrepresented very often in two ways. One is that mining itself is one of the few industries that is completely geographically independent. It doesn't matter where you are if you are mining, as long as you have an expensive electricity. What that allows you to do is choose the location of your mining system, based entirely on the local cost of electricity. That means that mining is doing market arbitrage for the cheapest sources of electricity. In many cases, the reason a source of electricity is cheap is because it does not match the demand at that location. In countries where there aren't broadly deployed, efficient distribution networks for electricity, one of the challenges with rapidly deploying and developing energy is that you build a power plant, not for the demand it has today, but for the demand you will have over the next 15 years. You have to make that investment now and grow into it, because you can't simply move energy across a massive country... when you don't have a distribution network. Speaking of China, if you didn't figure it out, that's one of the fundamental problems they have. This problem is replicated across not just China, but many developing nations as well as many developed nations. The places where electricity is needed, and the times when electricity is needed, are almost always not the places and times where electricity is available. What happens when you build a 50 megawatt plant in a place where they only have 15 megawatts of demand? In some cases, if it's alternative energy like wind, solar, or hydro, you can't turn it off or turn it down. You've built it, and it will produce, and then what? You're basically wasting energy. What if, in that environment, you could find a way to turn that energy into an alternative store value? Instead of paying off for the electricity plant in five years, you pay it off in one here, by using electricity that would be otherwise wasted. Now, Bitcoin is an environmental subsidy to alternative energy all around the world, because it's causing these projects to be amortized over a year instead of five. You're telling me we were running a green coin all this time, and I didn't even notice? One problem is that you've got to understand it from the perspective of balancing supply and demand on a global scale... without distribution networks. The decentralization of Bitcoin is driving the decentralization of energy production, which is one of the most important trends in human history. One of the other things you've got to keep in mind is that Bitcoin is easy to criticize for its energy consumption, because it's obvious. There are a lot more wasteful things that are far less obvious. For every time you pull out that little plastic card and use it to do a transaction, you're not aware of the 100,000 square foot data center that is churning 100,000 servers to do fraud detection, or clearing, or whatever. You're not aware of the tower offices that are lit 24 hours a day, and the trading floors, and the bank vaults, and the armored cars, and the diesel trucks, and the blah, blah, blah, blah. All of those costs are mostly hidden, and they're enormous. Are you telling me that perhaps Bitcoin ends up being the most efficient way to do transactions on a global scale? Remember that the level of security we have for Bitcoin today is a level of security that can handle global attacks by colluding nation-states. That's the level of security that is needed for this system to remain censorship-resistant. But if the system was ten times bigger with ten times more users, it doesn't need ten times more mining. It already has globally secure mining. What we have is enough. There's a profit motive that drives it. But it's a mistake to think that simply if we go global, that cost will also multiply. Quite the opposite, in fact. Over time, the reward from mining decreases. As a result, it's more likely we'll see that gradually taper off and plateau. This is a long game. The implications and complexity of how cost is allocated and how energy is consumed is huge. I don't think we can afford two proof-of-work systems on this planet, but I think we might only need one. Maybe everything else could be proof-of-stake and anchor into the only proof-of-work system we have. We need one planetary-scale proof-of-work system to offer us true energy-dependent immutability. But maybe we can only afford one. Turns out that might be Bitcoin's killer app.