3 Doors Down

Yay. There was a better turn out than expected. 4 votes! WOOHOO!

Hehe. Anyway, the results of the poll are 2-2. Now, time for the answer.

This is actually a pretty famous maths question known as the Monty Hall problem. It's a question of probability, and humans are inherently bad at questions of probability. This is because a lot of probability questions have answers which tend to be counter-intuitive. Anyhoo, let's get to the game.

The question is simple. We are given 3 doors. One door gives us a win. Two doors give a loss. We want a win. Our aim is to determine the best strategy to use in such a situation to win. Anyone with basic knowledge of probability should be able to follow.

Consider the twist given. After we choose a door, a losing door will be opened. We will then be left with the choice to change our choice or stick with it. Does it really matter?

One way to look at it is this. No matter what we choose, in the end, it will always be a choice between win or lose. So we can see this as 50-50. It doesn't really matter if we switch or not. So some might choose to stay. Some may choose to switch. It doesn't matter.

Here's the catch. DOES IT REALLY NOT MATTER? Maybe you've heard of the term "Conditional Probability". Maybe you know that probability can be very tricky. Maybe you think this is obviously a trick question, and change your mind on whatever your first instinct was. Whatever the case, here's the answer as given by statisticians.

Consider the game played without the twist. What would your chances be at winning? Quite obviously 1/3.

Now, consider playing the game with the twist. We are asked whether we want to change. If we have already made up our minds NEVER to change, then it doesn't make a difference whether we played with the twist or not. So it is as if we are playing the game without the twist. The chance of winning is 1/3. Follow?

Now consider this interesting fact which is often noticed but not appreciated fully. What are the chances that we lose when we choose from the start? Quite obviously, 2/3.

Applying the same logic from the above scenario, if we don't switch, our chances remain the same: 2/3 chance of losing, 1/3 chance of winning.

Now what if we always switch. Hehehe.

This stance assumes that we are wrong on our first choice. It is not always that we will be wrong, just 2/3 of the time. See where this goes?

If we start off by thinking we more likely to choose the wrong door at first, if we switch, it follows it is more likely to be the right door. The probability of 2/3 now comes into our favour, by turning it into the chance of winning.

In case you don't follow, this youtube video should do the trick.

Hope you had fun.