Monty Hall and Gambler’s Ruin: A Third Small Step into Mathematics

Today we will be looking at two problems in probability: the Monty Hall problem and Gambler’s Ruin. These are two common probability “brain teasers”. For the Monty Hall problem, it feels paradoxical when you first learn about it, while the Gambler’s Ruin is instructive and important as you go forward in life. With that brief introduction, let us begin!

There was a game show (so I have been told, though I am too young to have watched it) hosted by Monty Hall. One of the main elements of the show was a classic three-door setup, where behind two doors there were goats, and behind the third door there was a brand new car waiting to be won. The way this would go is that Monty Hall would ask you, the contestant, what door you wished to select. You picked door one, two or three. Then Monty Hall, bold and clever, would open up one of the two doors that you did not pick, only to reveal one of the goats! At this point in the game he would come back to you with a smirk, asking if you wished to stay with the door you picked at first, or if you wanted to switch to the other closed door. The problem is: in order to maximize the probability that you win the car, do you switch or stay (or does it even matter)?

Think about what your intuition says. Do you think it matters? Should you switch half the time and stay half the time? Always stay? Convince yourself of an answer and consider why you think it, then keep on reading.

I will not begin by giving you the answer, since that is no fun. We have to derive the answer from what we know. First, what is the probability that you picked the car correctly on your initial pick? 1/3 right? Definitely. So that is a good start. Now here is the more important part: What is the probability that you did NOT pick the car correctly? 2/3 of course. Now we get to the interesting part.

You are standing there, knowing that you only had a 1 in 3 chance of getting the car on your first pick; Monty Hall opens up one of the doors you did not pick and a goat walks out. How have the probabilities changed? This is the tricky part: they haven’t! You still have a 1/3 probability that you picked the correct door, and there is still a 2/3 probability that you did not pick the correct door. Just because a door was opened, this does not change. However, since a door was opened, you now know there is a 2/3 probability that the car is in the unopened door that you did not pick. Thus you should switch doors when asked.

Wait, what? Were you expecting that? It feels weird, because it is weird. Many people assume it doesn’t matter, since once the door is opened the probabilities switched to 1/2 for the car, 1/2 against. But that answer is obtained by switching the wrong probabilities. What is actually happening is that the 2/3 probability of the car being in the other two doors gets concentrated in the door you did not pick!

If you still don’t believe me, I have two more ways of explaining it: another example and a picture. Let’s start with the other example. Assume instead of 3 doors, we have 100 doors. Then the probability that you pick the car initially is 1/100. Now suppose that Monty Hall opens up 98 doors you did not pick, each having a goat behind them, leaving only your door and one other door. At this point it generally becomes more clear that there is not a 50-50 chance of getting the car. The probability that you picked the car correctly to start is so slim that you dramatically increase your chances of winning by switching. Now from my experience, there will still be many people who are not convinced by this argument, so let the following picture explain a bit.

This diagram lays out the problem very nicely. It displays how when you pick the car (situation 1) you will lose by switching, but if you switch in the other 2 cases you win. Thus 2 out of 3 times you win by switching. Keep looking at this, and eventually it will begin to click. 

One problem many people have is in situation 1. They see two options for you, depending on which door the host opens. However we do not count these separately, since they are indisitinguishable events, so whether the host reveals goat A or B, it is the same event.

This problem was much more difficult to grasp than our next topic, but it did help us think critically about probability and how to carefully navigate situations where chance is involved. This will be helpful as we talk about the Gambler’s Ruin. This is another classic in probability, which addresses how people in gambling games feel that a certain outcome is “due” after a long string of another outcome. 

Picture yourself at a simple casino game, where the employee residing over it flips a fair coin for you. In a roulette style betting, you can put money on either heads or tails. You sit at the table for a while, watching some other people play and seeing how everyone is doing. After a while, you notice that there have been 11 heads in a row. Feeling that the next one is very likely to be tails, you take a seat at the table and bet 5 times in a row on tails, only to be disappointed. There were now 16 heads in a row, and you feel cheated by the gods of chance. Surely this is improbable!

If you feel this way reading through this, let us have a little chat about independence of events in probability. Suppose I flip a coin. What is the probability I get heads? It’s 1/2 of course. It is also the same for tails. Now suppose that I just flipped and got a heads, and now I flip again. What is the new probability of getting a tails? Have the probabilities changed? Of course not! Each flip is independent of each other flip, and the prior outcomes do not affect the next toss.

Let us return to our casino. You realize there have been 11 heads in a row, which you feel is rather improbable, and knowing that over time the number of heads and tails should average out to be almost equal, you believe that tails is due. However, let me ask you this: Is it more probable to get 12 heads in a row, or 11 heads in a row followed by 1 tail? They are actually equal! The issue with the Gambler’s Ruin is two-fold: people try and use averages on independent events to determine the next outcome, and they also use aggregate probabilities instead of individual probabilities. We will look at each of these (potentially confusing) results.

First is the idea of averaging. We will again think about the coin example. After 11 heads in a row, you expect there to be a tails because “they need to average out”. Indeed, this is a valid interpretation of probability, that eventually outcomes in aggregate will have occurred according to their probabilities. One issue with applying this in our case is that 11 is a tiny number with regards to probability. We expect the results to average out as we approach infinity. Thus we need thousands or millions or billions of trials to be sure of averaging out. However, this still would not help, and this is simply because the events are independent. No matter what occurred before, we can consider each new flip of the coin (or roll of a die, deal of a deck, spin of a wheel) independent from the previous ones. The probability still remains the same.

The second issue is attempting to apply aggregate probabilities to individual events. What I referenced before is that the probability of obtaining 1 tail after 11 heads is exactly the same as obtaining 12 heads in a row. However, people will focus on the probability of obtaining 1 tail over the course of 12 flips, which is entirely different. Naturally it is more likely to obtain 1 tail over the course of 12 flips than obtaining all heads, but this is not relevant to the issue at hand. One should focus on the next event, and only the next event.

The Gambler’s Ruin is called such because it is a truly dangerous mindset to have when in a gambling situation. It applies all over the place: card games, the lottery, roulette, dice. The only place it may not apply as much is slot machines, but this is because there is a legal mandate about the winnings released so they do not only take money. So as you go forward, continue to think critically about probability and use the information in your decisions. It is extremely helpful and rewarding in the long run!

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