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?
Nope.
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
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. Â 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”
and
“that even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer.”
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. 🙂
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