There is no physics in this post, but there is some math. It is easy math though. It is the kind of math that can win you a big prize on a game show. Specifically, I am thinking about the game show Let’s Make a Deal. The original host of the show was named Monty Hall, and so this math problem is usually called the Monty Hall problem.
Often in the show an audience member is chosen and they are offered three doors to choose from. The contestant wins whatever is behind the door they choose. One of the doors hides a really great prize, like a brand new car, while two of the doors have a boring, stupid goat behind them. However, the interesting bit, and what the show is really about, is that just picking a door is not the end of the game. After the contestant picks a door, good old Monty shows them a different door, one of the two the contestant did not pick, that has a goat behind it. Then, he offers the contestant a choice: stick with the door they chose or switch to the other unopened door and win what is behind it. So, what option do you think is more likely to win you a new car? Many people, in fact 87% of people according to Wikipedia, stay with the original door they chose. However, I am going to try to convince you that you double your chances of winning the car if you switch doors.
There are several different ways to try to understand this problem that are explained in the Wikipedia article, but let’s first look at a slightly different situation. Imagine, instead of three doors, there are 1000 doors. When you first pick a door, there is a 1/1000 chance you picked a car. Then, Monty opens up door after door with goats behind them, until there is only one other door left unopened. Do you switch to that door? Of course you do! Monty just eliminated 998 wrong answers for you, so the odds that a care is behind that last door are quite good, 999/1000, much better than 1/1000. The same principle applies when there are only two doors. When you make your initial guess, you have a 1/3 chance of being correct. But when Monty shows you an incorrect answer, the odds that the unopened door is the correct one is 2/3, better than the original chance you guessed right.
Perhaps an easier way to see it is this. When you make your initial guess, there is a 1/3 chance the car is behind the door you chose and there is a 2/3 chance that the car is behind one of the other two doors. When Monty opens up the door with a goat behind it, there is a zero probability there is a car there. This does not change the probability that you guessed right though. There is still just a 1/3 chance you have the right door. That means that there is still a 2/3 chance the car is not behind the door you chose, and because Monty made one of those doors a zero probability, then the remaining door has all of that 2/3 probability. So, switching doubles your chances of winning a car!
The same is true for the game show Deal Or No Deal. There are 26 cases and you get to choose one, hoping that it has the grand prize. If you get to the point where there are just two cases left, you are given a chance to swap cases. Based on the above arguments you would be a fool not to switch.
So, if anybody reading this ends up on one of these shows and does not switch when given the opportunity, just remember that I tried to inform you and give you the best chance of winning it all. Good luck!