To Switch Or Not To Switch. Part 2

If math bores or offends you in any way, then please, by all means, ignore this article.

Other wise, please enjoy to what this article was to offer.

The rules are simple. The stakes are high. The anticipation. The Pressure. mm*mmph. It's unbelievably palpable. Before we get fried, you should probably know the rules: You are on a game show. There are three doors. Behind one door is a brand new car. Behind the other two doors are two "booby-prizes." You choose a door, which is not yet to be opened. Then the game host, knowing the content behind the doors, opens a door with a booby-prize behind it. You are given a chance to switch your choice. So, is it to your best interest to switch your choice? This is a problem dealing with probability. What are the chances of winning if you don't switch? What are the chances of winning if you do switch? At first, we are just completely confused, because this problem deals with probability within probability. The game host messes with your fundamental understanding of probability by showing you a "booby-prize."

At second thought, we apply Descartes' rules and turn the mind-boggling problem into a simple problem. We eliminate the booby-prize that the game host gives us. So now we are left with the car and another booby-prize. So, we have 2 choices; a 50/50 chance of winning. But is this the right approach? Should we leave the given booby-prize out of the equation?

In fact, we shouldn't. When we first choose a door, we have 1/3 chance of choosing the right door. And if we don't change our initial guess, the answer stays as a 1/3 likelihood of being the correct outcome. Notice: the probability does not change when the game host shows you a booby prize. If we switch our choice, everything changes. In fact, our chance of winning is 2 times a greater, 2/3 chance of winning, because unless our initial choice was the car, we'll win if we switch, giving us a 2/3 chance of winning.

Here's a visual that shows you all the possibilities:

Notice, I discreetly laid out all the possibilities. The chance of every choice is equally likely. The left side are choices with the outcome if we switched our initial guess. The right are the choices with the outcome if we did not switch our initial guess. The visual is extremely helpful. With this in mind, the first approach to a problem may not always be the correct one. In fact, it usually isn't. Only laying out a foolproof plan can we truly figure out the problem knowing we obtained the correct solution.

Here's a little computer stimulated graph that shows the experimental probabilistic experiment: http://upload.wikimedia.org/wikipedia/commons/0/0c/Monty_problem_monte_carlo.svg

So, what was your insight to the problem?

-idea contributed by Rocket (thanks!)