Probability Theory

Using Probabilities

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.

Resources

Books

Freund, John E., and Richard Smith. Statistics: A First Course. Englewood Cliffs, NJ: Prentice Hall Inc., 1986.

McGervey, John D. Probabilities in Everyday Life. New York: Ivy Books, 1986.

Perry Romanowski

KEY TERMS

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Combination

—A method of counting events in which order does not matter.

Conditional probabilities

—The chances of the occupance of an event given the occupance of a related second event.

Empirical approach

—A method for determining probabilities based on experimentation.

Event

—A set of occurrences which satisfy a desired condition.

Independent probabilities

—The chances of the occupance of one event is not affected by the occupance or non occupance of another event.

Law of large numbers

—A mathematical notion which states that as the number of trials of an empirical experiment increases, the frequency of an event divided by the total number of trials approaches the theoretical probability.

Mathematical expectation

—The average outcome anticipated when an experiment, or bet, is repeated a large number of times.

Mutually exclusive

—Refers to events which can not happen at the same time.

Outcomes

—The result of a single experiment trial.

Permutation

—Any arrangement of objects in a definite order.

Sample space

—The set of all possible outcomes for any experiment.

Theoretical approach

—A method of determining probabilities by mathematically calculating the number of times an event can occur.