Suppose you participate in a game in which you are to choose which among a set of containers holds a special item. To be a little more specific, suppose you have been admitted to a game show in which there are multiple boxes, and at least one box contains a huge amount of money while the rest contains nothing (or perhaps, some less exciting consolation prize). After choosing your box, the host then decides to tease the audience by opening one of the boxes you didn’t choose, only to find that it does not contain the cash (of course it doesn’t – if it does, then the show would be over!). You are then given the choice: will you stick with your box, or will you switch to a different box?

Naturally, you might think it’s all up to chance, and there really isn’t any advantage in whatever decision you choose. But surprise! There actually is. The reality is that you should switch boxes, because doing so will increase your chances of winning. In fact, if there are only three boxes to choose from, switching the box will double your chances of winning! This is known as the Monty Hall Problem.

The Math

To simplify the explanation behind this, let’s assume there are only three boxes to choose from, and only one of them contains the cash. Naturally, picking any one of those boxes will result in a 1/3 chance of getting the winning box. Once a box has been selected, there is now a 1/3 chance of the cash being in your box, and a 2/3 chance of it being in the other boxes. Now, when the host opens a box to show that it is empty, these values don’t really change: there’s still a 1/3 chance that the cash is in your box, and there’s still a 2/3 chance that the cash is in one of the other two boxes. However, by revealing the empty box, the host really lets you know that there’s 0% chance of the cash being in that box: So, this means there’s a 2/3 chance of the cash being inside the only other remaining box that the host did not open! So by switching your box, you increase your chances from 1/3 to 2/3, meaning you actually double your chances. Of course, if the are more boxes to choose from, the increase in your chances becomes less pronounced. For example, if there are four boxes to choose from, you can do the math similarly to show that switching will increase your chances from 1/4 (25%) to 3/8 (around 38%). Not much, but you increase your chances nevertheless.

Of course, it’s all still a game of chances, so switching boxes will not guarantee you a win. However, in such games of chances, increasing your chances will never hurt. So the next time you are faced with such a situation, always switch your boxes!

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