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## Introduction to Probability

When you flip a coin, you experience probability. The outcome, either heads or tails, is for you random, with a one-half chance of either.

But is this event really random? If a physicist has all the information on spin, force of the toss, and drop distance, he might be able to predict the outcome. There are some people who have practiced tossing coins so that they can largely determine whether heads or tails comes up. The very notion of randomness posses interesting philosophical questions that are no concern of ours. We will assume any such problems away because randomness and probability are extremely useful ideas that, somewhat unexpectedly, help us control our world.

Mathematicians initially developed the theory of probability to explain games of chance. Probability is a number between zero and one attached to possible outcomes. A probability of zero means that the outcome is impossible and a probability of one means that the outcome is certain to happen.

The probability that an event will not happen is one minus the probability that it will happen, or in mathematical notation:

Prob(Not A) = 1 - Prob(A)

This simple notion helps explain a result that some people struggle to understand, the Monty Hall Problem. Suppose you are asked to choose one of three doors. Behind one of the doors is a valuable prize, and nothing is behind the other two. After you choose, Monty, the person running the game and who knows where the valuable prize is located, opens one of the doors that you did not choose and says, "Aren't you glad you did not choose this door because there is nothing here. Would you like to stay with your original choice, or would you like to switch to the other closed door?"

What would you do? Most people stick with their original choice, thinking that they now have a 50-50 chance of winning. But they are wrong. If they could play this game an unlimited number of times, they would win 1/3 of the time by sticking with their original choice. If the probability of winning by keeping the original door is 1/3, what does the little formula above tell you about the probability of switching?

(There are several on-line simulations of the Monty Hall Problem that will let you play the game over and over, and using it you can verify that the correct strategy is to switch. See, for example, here. A much more elaborate explanation is here.)

It may be worth noting that people have a hard time with the notion of randomness. Our brains are programmed to find order and make sense of things, so we have a tendency to see patterns even when there are none. Also, people want to think that they have some control even when they do not. Hence, lotteries draw many more participants when people can pick their numbers rather than have them randomly assigned even though the odds of winning are identical in both cases. Where lotteries are popular, schemes for picking numbers are popular, implying that people do not see the situation as purely random. The illusion of control was frequently on display on the game show, "Deal or No Deal," where the contestants and the moderator often treated the picking of numbers as a matter of skill when no skill was involved.

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