Probability Theory

Probability theory was originally developed to help gamblers determine the best bet to make in a given situation. Suppose a gambler had a choice between two bets; she could either wager $4 on a coin toss in which she would make $8 if it came up heads or she could bet $4 on the roll of a die and make $8 if it lands on a 6. By using the idea of mathematical expectation she could determine which is the better bet. Mathematical expectation is defined as the average outcome anticipated when an experiment, or bet, is repeated a large number of times. In its simplest form, it is equal to the product of the amount a player stands to win and the probability of the event. In our example, the gambler will expect to win $8 × 0.5 = $4 on the coin flip and $8 × 0.17 = $1.33 on the roll of the die. Since the expectation is higher for the coin toss, this bet is better.

When more than one winning combination is possible, the expectation is equal to the sum of the individual expectations. Consider the situation in which a person can purchase one of 500 lottery tickets where first prize is $1000 and second prize is $500. In this case, his or her expectation is $1000 × (1/500) + $500 × (1/500) = $3. This means that if the same lottery was repeated many times, one would expect to win an average of $3 on every ticket purchased.