One of the fun probability mind-bogglers of our time is that of a game show. Imagine you are on a game show and you are presented 3 curtains. Behind one is a brand new car, behind the others are goats. Obviously you'd want the car. So you chose one of the three curtains. Now obviously the probability you chose right is 1/3 or P(1/3). But if this game show is anything like Monte Hall's, after you chose your curtain they reveal one of the curtains WITH a goat behind it. Now they give you the option to switch. So the question is do you have a higher chance of winning the car if you switch, or if you stick to your guns.
Some people will say it doesn't matter, it just becomes a 50-50 chance right? Well not exactly. So lets assume that you a a contestant will never switch. If you pick curtain #1, and then Monte Hall (your game show host) shows that curtain number three has a goat too, you still have the same chance of winning even if he hadn't shown the curtain, 1/3. Now if a mathematician comes on to the show, this is how he would approach it. The probability of getting the first guess wrong, is 2/3, and if you get the first guess wrong (choose a goat) and then switch, you will always get the car. This is because obviously you're not going to choose the revealed goat curtain. So, you want to get a goat on the first guess, so you can see where the other goat is, and then switch to the car. The probability of getting the car if you always switch becomes 2/3.
So in conclusion, always switch on those game shows. Your probability increases from 1/3 to 2/3's if you do.
