Archive for the ‘rational thinking’ Category

Math is probably for you

November 22, 2009

Math can be really fun. Seriously.

This post is the 2nd in a series of posts I’m planning to have about why math is such a beautiful, useful, and awe-inspiring subject, and that a lot of us can do math (advanced/seemingly difficult math even). Math is such an integral part of humanity since our cave dwelling days, and much more so now in most of our technology driven lives. Previously I wrote about how even advanced math, particularly advanced geometry, can be easily tackled with just your imagination. This time it’s about probability. I can just imagine some of you cringe at the thought of math, let alone probability. But I’ll try to show you that often times, logical reasoning is all that it takes to wrap your head around probabilities, even the ones that confound a lot of brilliant people, even some mathematicians themselves. In fact, we’ll end this article with a simulation of a game/game show. Not bad huh? 🙂

Probability and people

In a nutshell, probability is the area of math which deals with the likelihood of an event happening. It is usually expressed as a number, whether a fraction or a decimal, between 0 and 1, with a probability of 1 meaning the event will surely happen and a probability of 0 meaning the event won’t happen.

Now, don’t be too hard on yourself thinking that probability is too hard for you, unlike most of the human population. In fact, probability is one really confounding area of math and problems in it that seem to be easy in hindsight, turn out to be deceptively difficult or tricky, even for  mathematicians, teachers, and other brilliant men and women around the globe. In fact a lot of us have trouble wrapping our heads around probabilities. You mix that with human hopefulness and also the difficulty of grasping very large numbers and what you get is the staggering number of people around the world falling in line to get their lotto tickets so they could win the multi-million prize money.

In fact, if we do the math, in a typical 6/49 game of lotto (6 unique numbers chosen out of 49 numbers, where the order of the 6 numbers is not important) we find that your chances of winning today after buying that lotto ticket is 1 in about 14,000,000. So if Lucy (one of the earliest hominids/proto-humans known to us) or her people, or perhaps even Neanderthals started betting on the lottery at the beginning of their lives, some of them should be millionaires by now. That’s how bad we are at assessing odds, especially coupled with large numbers. So when you go buy that lotto ticket later, I’m afraid the odds are so much against you.

However, I’ll discuss next a particularly perplexing probability problem pondered by people, even brilliant ones, and found the solution to be deceptively trivial after all. Actually, even after you get the explanation, from a practical standpoint it doesn’t seem like so. But the logical reasoning will quite surely buy you out. But don’t fret, all you need again is imagination and logical reasoning. 🙂

Game time

Some of you may have heard/read about the American game show Let’s Make a Deal. The Monty Hall problem (MHP) was named after the show’s host. Simply stated, the rules of the game are as follows:

The game master (GM), has 3 doors: 2 with goats behind them and one with a car behind it. The GM lets you choose one door, which you think holds the prize car behind it. Since the GM’s job is to make you and the audience excited and enjoy the game, the GM opens another door. But since the GM knows the placement of the goats and the car i.e. which door has which item behind it, the GM opens a door which has a goat behind it. Now, the GM poses a question to you: Do you or do you not want to change the door you initially picked i.e. the GM gives you an opportunity to stay with the door you originally picked, or to choose the other door, knowing that one of the doors, which the GM opened, has a goat behind it.

The GM in the show is of course Monty Hall (MH). Now, you’d most probably think that since there are only 2 doors left unopened, that the probability of getting either a goat or a car is now 50/50 or 50% right i.e. it doesn’t matter whether you switch doors or not?


In fact, however counterintuitive this may seem, your chances of getting the car at this point of the game doubles if you decide to change the door you initially picked. How? Let’s find out shall we? 🙂

Goat, Car, Goat

Now let’s strap on our imagination and logical reasoning caps to find out how the probability of getting the car increases two-fold if you switch your chosen door, and that it’s not a 50/50 chance of getting the car once a door with a goat has been opened by the GM.

Monty Hall problem

Monty Hall problem

One way of looking at how this counterintuitive probability problem is correctly tackled is by taking the possibility of the events one at a time (refer to the figure above please). In this scenario we show that when you switch doors, you always double your chances of winning. Here’s how:

1. First event, say you picked a door and it happened to have the prize car behind it.  Regardless of which door the GM opens, switching in this case either gives you goat A or goat B i.e. you lose the prize car. Out of the 3 possible scenarios (2 of which are listed right after this one), in this one event/case do you lose the prize car.

2. Second event, you choose a door with a goat (goat A) behind it. The GM opens a door again with a goat (goat B) behind it. If you switch in this case, you get the car. This event, wherein you get the car by switching, is one event which you get the prize car. Score one for you. 🙂

3. Third event, you choose the 3rd door with a goat(goat B) behind it. The GM again opens a door with a goat (this time, goat A). So when you switch, you get the car again. Yay. 🙂 This event, wherein you again get the car by switching, is another event which lets you take home the prize.

So what did we get from all this? We saw that out of 3 events/cases of picking either of the 3 doors, you always get 2 events (event 2. and 3.) which favor switching and which lets you walk away with the prize (or in this case, drive away with the prize). So the odds of getting the car/prize in the MHP is not 50% as a lot of us would initially assume, but instead, is really 2/3 or approximately 66.7%.

It can take a while to sink in, but the reasoning/explanation is quite logical and sound.

Try it out!

I actually tried this out with my mother and at another time with my younger brother. What I did was I got 3 opaque plastic cups (simulating the doors) and 2 toy cows (no goat toys in our house at that time) and 1 robot toy that transforms into a car (not bad for a prize no?). I made them act as a GM at one time, with me being the game contestant. Of course to prove my point I always switched. We did this about 20 times and I got the prize car (or robot) at around 14 times out of the 20 (roughly 2/3 of 20). Then I acted as a GM and they acted as the contestant. Then their job was not to switch doors (or cups), just to prove my point that you get the prize more often than not (2/3 of the time remember?) by switching instead of staying with your original door/cup.

They even asked me if I was doing a magic trick on them. I told them it was the power of mathematics and of logical thinking. 🙂 Imagine what much more primitive, let’s say Bronze-aged men, would think of me, with this knowledge, even without modern devices like a cellphone. Perhaps they’d think of me as an oracle or even a god. 🙂

Great, great. But what’s the use?

I think one important thing we can get from this (other than to show you that you can do maths you thought were too hard or complicated for you) is that with math, we can make decisions in our lives (sports betting, lottery, game shows and so on) with more clarity, logic, and sound reasoning, instead of just blind optimism.

If you didn’t get the logic on how to win the game at first glance, or if you thought it was 50/50, don’t be ashamed, a lot of people (some brilliant even) fell for it too. In fact, out of 228 subjects in a study, only 13% chose to switch, and that the rest (87%) assumed that the switching didn’t matter since the likelihood of getting the car out of the 2 unopened doors are equal (research by Mueser and Granberg, 1999).

Quoting cognitive psychologist Massimo Piattelli-Palmarini

“… no other statistical puzzle comes so close to fooling all the people all the time”



So, not bad eh? Still think math (or at least those areas you think are too advanced or complicated for you) isn’t for the average person? If so, then look forward to my next posts about math. 🙂

References, resources, and further reading

Gather ’round kids, it’s time for math!

November 1, 2009

Mathematics is for everyone. Really.

This article ( and the succeeding ones in the series) aims to prove that point. That everyone has a mathematical brain. Specifically, I’ll concentrate on a certain area of mathematics in this article known as geometry, and then go to more advanced geometry (usually college or graduate level geometry). Don’t fret! There are no equations here which will make your eyes wander and do something else (at least while you’re reading the article). There are a lot of  science articles around, but what you usually don’t get often are articles about math, how beautiful and useful it is, and how important it is to science and modern civilization.


Towards healthier skepticism: Correlation does not imply causation

October 9, 2009

This post will attempt to repeat, clarify, and elucidate the need for the remembrance and understanding of the phrase “correlation does not imply causation”. Scientific studies will be given, and the words in the phrase, which vary in meaning depending on usage, will be defined accordingly.

Scientific studies

Please take a moment to go through the following actual, summarized scientific research results:

1) In a previous scientific research using quantitative assessment, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD.

2) From a study at the University of Pennsylvania Medical Center, young children who sleep with the light on are much more likely to develop myopia in later life.

We will get back to them in a moment. Now we focus on correlation or co-relation, and why scientists, statisticians and skeptics, at the very least, should always maintain and promote the phrase “Correlation does not imply causation”.


Estimating Distances Technique – A Detailed Inspection

September 14, 2009

This post is about a web article which mentions a technique that allows you to estimate, to a relatively good degree, your distance from another object. I’ll then explain the minor error which the article has, as well as the assumptions of the technique which the web article did not mention.

The Technique

I came across this neat trick from lifehacker.com about estimating distances using your arm and thumb. It is quite useful, but below I will outline a relatively minor error of the article, particularly the diagram used. It was a minor error but it still strikes me as something that should be brought to light, since it’s pretty trivial too. The original article btw was taken from the almanac.com article of the same name. Both articles were quite short so they only took me a small amount of time to read through them and to quickly notice that there was something wrong with the diagram.

The articles emphasize the fact that one’s arm (held straight) is approximately 10 times longer than the distance between your eyes. The articles also mention that with a bit of applied trigonometry, one can estimate distances between you and an object which you have a reliable width knowledge of. Unfortunately the article writer/s might have focused on the trigonometry part too much, overlooking their basic geometry when they created the diagram.

Here is the original almanac.com diagram showing a man estimating his distance from a barn which he originally knows the approximate width.


How The Technique Works

In case you haven’t read the original article yet, it basically says that (again from almanac.com):

  • Hold one arm straight out in front of you, elbow straight, thumb pointing up.

  • Close one eye, and align one edge of your thumb with one edge of the barn.

  • Without moving your head or arm, switch eyes, now sighting with the eye that was closed and closing the other.

  • Your thumb will appear to jump sideways as a result of the change in perspective.

How far did it move? (Be sure to sight the same edge of your thumb when you switch eyes.)

  • Let’s say it jumped about five times the width of the barn, or about 500 feet.

  • Now multiply that figure by the handy constant 10 (the ratio of the length of your arm to the distance between your eyes).

Now you get the distance between you and the barn—5,000 feet, or about one mile. The accompanying diagram should make the whole process clear (shown above).

The Error In The Original Diagram

The error comes from the fact that the original diagram, whether it be the vertical one from almanac.com or the horizontal, modified version from lifehacker.com, show the distance line not being parallel to one line common to both triangles formed. To see this more clearly, I’ve created a little more technical and descriptive diagram below. The new diagram shows, correctly, that the distance line (containing the 5000′ and 20” distance markings) is parallel to the line connecting the observer’s left eye to the barn’s new location. That’s it. That’s the error 🙂 It may seem trivial, and it actually is, but I couldn’t help noticing it, especially since apparently no one has commented about it, and some people I know who should have noticed it, didn’t. 🙂 The original diagram shows the left-eye-new-barn-location line to be non-parallel to the distance line, which is wrong, and which quickly caught my skeptical eye. Basic geometry will tell you that my new diagram below is the more correct one.

wp-blog-post-estimating distances 2009-09-13

Assumptions Which Were Left Out

The assumptions which the article does not mention include:

  1. One knows a relatively precise measurement of the object’s width, or that one should know a good deal about the object’s width before attempting to estimate distance with this technique. To see how this can become a problem if not entirely taken into consideration, suppose you estimated or falsely remembered that the barn was 400ft instead of 500ft. That would translate to your estimated distance of 4000ft, which is 1000ft shorter than the correct 5000ft! 🙂 You’d then get a nasty surprise since you left out 1000ft. In other words, since the ratio of the object’s width to the distance between it and you is 10, your width estimation errors (again, could be from wrong estimation or remembrance of the object’s width) get translated to a distance error multiplied by 10.
  2. The topography of the terrain. This technique assumes or works best in a plain, since if you were say in a hilly or mountainous region, the distance you’ll get from this technique is the straight line distance from you to the object. But it does not take into consideration the slope, nor the crests or troughs of the land. You may get a distance of 1000ft between you and the object, but if there are hills and such between you and the object, you know it will be more than 1000ft. 🙂

Gracias a mis amigos Rudolf y Aaron. Thanks to my friends Rudolf and Aaron for their quick help in confirming this error, since I wanted to be triply sure. 🙂

Quite Quotable Quotes: The Big Bang Theory

July 29, 2009

The Big Bang Theory (TBBT) is undoubtedly my favorite sitcom so far. I’ve never really been into sitcoms actually. Some of the last few ones I watched were (believe it or not) Seinfeld, Fraser, and Friends, and I didn’t really get into them that much. I just watched a few episodes here and there, usually with my dad or with my sister when we were much younger. TBBT has fervently rekindled my attention towards sitcoms, in such a magnitude I can only describe as the energy needed to accelerate an electron to 0.99% the speed of light 🙂

Needless to say, there are quite a lot of sources on the Internet for what TBBT is all about. Wikipedia or a simple Google search or a quick visit to the official site should do fine for a start. What it is to me however, is a brilliant show that combines geeks, nerds, comic books, sci-fi, technology, physics , science, and jokes together, and still be absolutely entertaining and humorous. In other words, much as what the Gay Liberation has done to reinvigorate gay pride, TBBT has reinvigorated the geek pride in me. The writers and producers are themselves geeks and nerds, watch Star Trek and Battlestar Galactica and read comic books. But they also treat the characters in the show with such respect that even if you’re not one of “them” (or in this case, one of “us”), you’d still find respect for them (or “us”). TBBT has I think, no doubt inspired many reluctant geeks and nerds, not just in America but across the globe where TBBT is being shown, to go out and be really proud to be geeks.

Without further ado, here are some of my favorite quotes from the first season:

From the season 1 Pilot episode:

Leonard: We need to widen our circle.
Leonard: Yes, and you’ve never met one of them.
Sheldon: That’s the beauty of it!

And yet another from the same episode:

Penny: I’m a Sagittarius, which probably tells you way more than you need to know.
Sheldon: Yes, it tells us that you participate in the mass cultural delusion that the sun’s apparent position relative to arbitrarily defined constellations at the time of your birth somehow affects your personality.
Penny: (puzzled) Participate in the what?

And another:

Sheldon: Okay, look, I think you have as much of a chance of having a sexual relationship with Penny as the Hubble telescope does of discovering that at the center of every black hole is a little man with a flashlight searching for a circuit breaker. Nevertheless, I do feel obligated to point out to you that she did not reject you. You did not ask her out.


Leonard: (talking about him and Penny) Our children will be smart and beautiful.

Sheldon: Not to mention imaginary.

And from the succeeding episodes:

Sheldon: You have to check your messages, Leonard! Leaving a message is one-half of a social contract, which is completed by the checking of the message. If that contract breaks down, then all social contracts break down and we descend into anarchy.
Leonard: It must be hell inside your head.
Sheldon: At times.

Wolowitz: If it’s “creepy” to use the Internet, military satellites, and robot aircraft to find a house full of gorgeous young models so I can drop in on them unexpected, then FINE, I’m “creepy”.


Sci-fi ponderings when (almost) idle

June 22, 2009

I was talking to one of my housemates last night while I was taking a break from using my PC when we came to the topic of Terminators.  Being sci-fi fans (me especially) we talked about the realism of robots and computer software taking over the world.

The proposed problems pertaining to plausibilities

Now, my housemate proposed that Skynet, the autonomous computer program that took over the world and is responsible for the decline and domination of the human race, isn’t very plausible, or at least isn’t ‘too’ smart. His propositions are the following:

a) Skynet should have made the Terminators smarter so as to make them more adaptable to human circumstances, issues, and environment.

He mentions that though they look like humans, they (or at least the ones in the movies, unlike in the Terminator TV series) they still act relatively cold and robot-like. Adapting to human behavior, emotions, idiosyncrasies, at least temporarily, may help them perform their missions better, i.e. terminating their targets.

b) Skynet, at least in the future (so my housemate concedes), should have made itself connected/linked to Terminators so that it can use it’s powerful processors and information on humans, including their tactics, to finally wipe out the human race and leave nothing to stand up against it.

I made my housemate actually concede early in this proposition that this can only work in the future, because how would Skynet of the future control and communicate with Terminators it has sent to the past? It could be quite given that future Skynet would be linked to the Terminator pawns via some wireless technology, but wireless technology across time? Dubious.

The proposed answers to the problems

a) Now this one has been answered already in Terminator 2: Judgement Day, when John Connor asks (approximately at 1 hour 6 minutes of the Special Edition of the movie) if Terminators (or at least the T-800 model 101 Terminator a.k.a. Arnold model/line of terminators) can learn new things so they can be more human. The terminator responds by saying that Skynet “presets the switch to ‘read-only’ when terminators are sent out alone”, to prevent them from “thinking too much”. This then prevents terminators by default, or at least the movie terminators such as the T-800, T-850 T-1000, and T-X, from learning a lot of things about what makes humans humans.

b) One solution I’ve thought about for this specific conundrum in the Terminator universe, which could also be said in real life hardware/software,  is that if Skynet ‘hooks’ itself up to every terminator walking around trying to find, infiltrate, and terminate humans, i.e. connect its thinking to the terminators, then that would lead to a vulnerability. The vulnerability comes from the fact that by doing so (hooking up/connecting to terminators in the field) would allow humans to insert a virus or ‘anti-Sknet’ software to one or more captured terminators, which could then be uploaded to the main Skynet program and destroy Skynet entirely. This is possible because Skynet has to maintain a duplex connection to the terminators in the field if Skynet is to control them and still be in sync with the main Skynet program. I think this is a risk Skynet would not dare take.

Questions/comments/arguments? Feel free to post them as long as they’re calm, ruly. 🙂


March 4, 2009

Mother, you bid to bathe me with boon, and still you do

Try, for you feel it’s your responsibility to.

You’ve sacrificed, and supplied me with sustenance

There was a time, I remember, when our parlance

Were very much attuned to each other’s ideas and thoughts

But then I studied and grew, and I think you see it was not for nought.


Mother, you never really showered us such spirituality as

Father did; he would try to preach the Bible from start till last.

Yet still you required religious reverence, relished it if you will

You believed in heaven and hell, Adam and Eve, and of course good and evil

As a child, I believed those, and I still remember believing them too

I was a victim of it all, indoctrination, and I think so were you


Mother of my mother, I’ve no doubt you’ve assiduously applied all

Your talent and time to transform this future mother’s mind into a thrall

Do I then blame you for bequeathing this bane? This irrational virus of the mind?

Perhaps so, but you are just a victim as well, in a long chain left behind

By forebears who were perhaps too quick to believe and be converted

From a father and a mother to their sons and daughters, the passing of the virus repeated.


Mother of my mother, father of my mother, and many of our ancestors

They took pleasure, and so do you I believe, from knowing your errors

Your actions, your thoughts, somebody is watching; Pretty silly if you think about it for a moment

That you take as a rule, Bronze age ideology, instead of actual discernment

Of how things really work, science showing the systematic silk-road to awareness, to evidence

Which luckily, some of our ancestors started, instead of balking in gods and their supposed omniscience


Motherland, you’ve spuriously spawned spirit satiated sapiens since stone ages

People who cling to quick reaffirmations, even if evidence says the erroneous explanation engages

Not man’s faculty of reason, but his irrationality and will to believe

And if it be so, would it not be possible then, that any lesser man contrive and conceive

Any ludicrous and fanciful mythology, unto his own machinations and whims?

“Of course!” That’s the answer. As Dawkins puts it, those inventions pass on as memes.


Motherland, how long will ‘heathens’ like me, unto which science and skepticism prevails,

Be beset by bothersome brigands banding together, to remove me from freethinking land from where I hail?

The time of rationality, of science, of constructive skepticism, of evidence seeking, should fill the 21st century

Yet the century is still plagued by, unfortunately, blind and unyielding irrationality

If what religion means is to be awed by our fleeting lives in the vast cosmos, then I’m a religious man

But I would not immediately jump blindly by mere faith, and always reason out if I can.

Shakespeare and programming

December 10, 2008

A great playwright and poet once wrote in his play Hamlet (and that great poet and playwright of course is none other than Shakespeare)  the following question from act three, scene one:

To be or not to be, that is the question;

Putting it into a more geeky format I have the following translation:


Which sort of turns the question into a logical statement. Tidying it up a little further and noting the unary and binary operators in the statement, as well as the operator precedence, and further clarifying its (geeky) nature I have:

0x2B OR (NOT 0x2B)

And so I arrive at an answer to the question in Hamlet’s soliloquy:

0x2B OR (NOT 0x2B) = 0xFF

The answer turns out to be pretty simple and not so philosophical and deep! 😀 If you don’t know why my answer to the famous question is 0xFF, keep on guessing! 😀

(I’m feeling geekier than usual tonight, so there you go)

There Is No God (And You Know It)

November 14, 2008

This is an article written by Sam Harris some time ago. I just reposted it here with permission from machineslikeus.com (thanks Norm). I reposted it in my web blog because it’s just light reading (i.e. no deep and highfalutin words), but it still delivers a very intense and profound idea: There is no God and you know it. Read it and find out why, and then perhaps leave a message why you agree/disagree.

There is No God (And You Know It)

by Sam Harris

Somewhere in the world a man has abducted a little girl. Soon he will rape, torture, and kill her. If an atrocity of this kind not occurring at precisely this moment, it will happen in a few hours, or days at most. Such is the confidence we can draw from the statistical laws that govern the lives of six billion human beings. The same statistics also suggest that this girl’s parents believe – at this very moment – that an all-powerful and all-loving God is watching over them and their family. Are they right to believe this? Is it good that they believe this?



It is worth noting that no one ever need identify himself as a non-astrologer or a non-alchemist. Consequently, we do not have words for people who deny the validity of these pseudo-disciplines. Likewise, “atheism” is a term that should not even exist. Atheism is nothing more than the noises reasonable people make when in the presence of religious dogma. The atheist is merely a person who believes that the 260 million Americans (eighty-seven percent of the population) who claim to “never doubt the existence of God” should be obliged to present evidence for his existence – and, indeed, for his benevolence, given the relentless destruction of innocent human beings we witness in the world each day. Only the atheist appreciates just how uncanny our situation is: most of us believe in a God that is every bit as specious as the gods of Mount Olympus; no person, whatever his or her qualifications, can seek public office in the United States without pretending to be certain that such a God exists; and much of what passes for public policy in our country conforms to religious taboos and superstitions appropriate to a medieval theocracy. Our circumstance is abject, indefensible, and terrifying. It would be hilarious if the stakes were not so high.

Consider: the city of New Orleans was recently destroyed by hurricane Katrina. At least a thousand people died, tens of thousands lost all their earthly possessions, and over a million have been displaced. It is safe to say that almost every person living in New Orleans at the moment Katrina struck believed in an omnipotent, omniscient, and compassionate God. But what was God doing while a hurricane laid waste to their city? Surely He heard the prayers of those elderly men and women who fled the rising waters for the safety of their attics, only to be slowly drowned there. These were people of faith. These were good men and women who had prayed throughout their lives. Only the atheist has the courage to admit the obvious: these poor people spent their lives in the company of an imaginary friend.

Of course, there had been ample warning that a storm “of biblical proportions” would strike New Orleans, and the human response to the ensuing disaster was tragically inept. But it was inept only by the light of science. Advance warning of Katrina’s path was wrested from mute Nature by meteorological calculations and satellite imagery. God told no one of his plans. Had the residents of New Orleans been content to rely on the beneficence of the Lord, they wouldn’t have known that a killer hurricane was bearing down upon them until they felt the first gusts of wind on their faces. And yet, a poll conducted by The Washington Post found that eighty percent of Katrina’s survivors claim that the event has only strengthened their faith in God.

As hurricane Katrina was devouring New Orleans, nearly a thousand Shiite pilgrims were trampled to death on a bridge in Iraq. There can be no doubt that these pilgrims believed mightily in the God of the Koran. Indeed, their lives were organized around the indisputable fact of his existence: their women walked veiled before him; their men regularly murdered one another over rival interpretations of his word. It would be remarkable if a single survivor of this tragedy lost his faith. More likely, the survivors imagine that they were spared through God’s grace.




Originally posted here.

Also, click here for Sam Harris’ official website.

The Woman I Shall Marry :D

November 9, 2008

Click here for the source.