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
The probability that an event will not happen is one
minus the probability that it will happen, or in
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
(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.