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# Probability, Theoretical

The theoretical probability of an event is based on the assumption that each of a number of possible outcomes is equally likely. The theoretical probability of an event can be defined as the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. For example, if a die is tossed, the probability of getting a 4 is because 4 is one of six possible outcomes. Similarly, the probability of getting a number less than 5 is , because either 1, 2, 3, or 4 is favorable.

The possible values of the theoretical probability of an event range from 0 to 1. If none of the possible outcomes is favorable to an event, the theoretical probability is 0. For example, the probability that the number attained on a roll of a die is greater than 8 is 0, because such an event is not a possible outcome. Similarly, the probability that the number attained is positive equals 1, because every possible outcome is favorable to this event.

## Understanding Equal Likelihood

The assumption of equally likely outcomes, although critical to the determination of theoretical probability, can also be somewhat misleading. For example, consider the person who, when asked what he thinks his chances are of winning the lottery, responds, "Fifty percent, because there are only two possibilities: either I will win or I will lose." Clearly in this situation, winning and losing are not equally likely!

Consider the sum that results when two dice are tossed. The possible outcomes for this experiment are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. These outcomes are not equally likely because they are based on a two-part experiment rather than a simple experiment. Yet the principle of equally likely outcomes can be applied to determine the appropriate individual theoretical probabilities.

To simplify the analysis, assume that one die is blue and the other red. The number attained on one die will not affect the number attained on the other. Thus, if a 1 is tossed on the blue die, the red die can show any of the numbers from 1 to 6. Similarly, if the blue die is 2, the red die can be any number from 1 to 6. Continuing in this fashion, one can see that the sample space for this experiment consists of thirty-six possible outcomes, any one of which is equally likely. Using the information from the matrix below, it can be determined that the probability of the sum of the dice being 10 is , or , since a sum of 10 occurs in three of the thirty-six possible outcomes.

## Games Involving Probability

Analyzing theoretical probabilities provides the basis for determining the best strategy to use in a wide range of games. It is important to keep in mind, however, that when games involving probabilistic events are played, the actual results will vary from the theoretical probabilities. In the short run, these differences may be quite dramatic.

For example, in the game of Monopoly, a player who lands in jail can either pay a fine or attempt to get out by rolling a pair of dice. If doubles are rolled (the same number appearing on each die), the player gets out of jail. Based on the matrix above, the theoretical probability of rolling doubles is ,or . But anyone who has played Monopoly knows that sometimes a player may roll the dice ten or more times before gaining freedom, whereas on other occasions a player gets out on the first roll.

In some situations, the determination of theoretical probability must be adjusted based on known information. In the seven-card stud version of poker, for example, each player receives seven cards, four of which are dealt face-up. Suppose a player's first six cards are the 3 of hearts, 6 of diamonds, 7 of diamonds, 9 of spades, jack of diamonds, and queen of diamonds. If the next card is a diamond, the player will have a flush (five cards of the same suit) and will most likely win, otherwise the player will almost certainly lose.

If the decision to remain in the game or drop out is based on the assumption that the probability of getting a diamond is 1/4 (because thirteen of the fifty-two cards in the deck are diamonds and equals ¼,) the reasoning will be seriously flawed. Instead, the player must consider all of the cards that have been seen.

If six other players are in the game, their twenty-four face-up cards will be known in addition to the six cards held by the player. If none of the other twenty-four cards seen is a diamond, the player's probability of getting the flush is , because it is equally likely that any of the twenty-two unseen cards will be dealt and nine of them are diamonds. But if nine of the twenty-four face-up cards of the other players are diamonds, the probability of getting the flush is 0, since all thirteen diamonds have already been dealt. Depending on the composition of the cards already seen, the probability of getting the flush may be significantly less than or greater than ¼.

see also Games; Gaming; Probability and the law of large numbers; Probability, Experimental.

Robert J. Quinn

## Bibliography

Campbell, Stephen K. Flaws and Fallacies in Statistical Thinking. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1998.

## LET'S FLIP FOR IT

The concept of theoretical probability is often used when the goal is to make a fair selection. The National Football League, for example, uses the flip of a coin to determine which of the two teams will have their choice of kicking or receiving when the game begins. In this way, neither team is given an advantage since the theoretical probability of heads is