The experimental probability of an event is defined as the ratio of the number of times the event occurs to the total number of trials performed. As a consequence of this definition, experimental probability is not a fixed value but varies depending on the particular set of trials on which it is based.
As with theoretical probability, the possible values of the experimental probability range from 0 to 1. An event would have an experimental probability of 0 if, during the trials performed, that outcome never occurred. But unlike theoretical probability, experimental probability bases its value solely on the results of the experiment, rather than assuming that certain events are equally likely to occur.
Performing experiments for the purposes of determining experimental probabilities can provide powerful insights into the nature of probability. Rolling dice, tossing coins, drawing marbles from a hat, and selecting playing cards from a deck are just a few of the common probability experiments that can be performed.
Consider, for example, the toss of a die. A die is a cube, each face of which displays from 1 to 6 dots. In tossing a die one hundred times, the following results were attained:
2, 4, 4, 5, 2, 3, 1, 5, 1, 3, 2, 1, 3, 3, 4, 5, 1, 5, 5, 4, 3, 6, 5, 6, 6, 5, 6, 6, 2, 6, 4, 4, 4, 3, 3, 2, 2, 4, 3, 4, 1, 2, 2, 5, 5, 6, 4, 1, 1, 6, 4, 2, 6, 6, 5, 1, 3, 4, 6, 1, 5, 4, 3, 1, 6, 4, 3, 5, 6, 2, 4, 6, 1, 5, 5, 3, 3, 6, 4, 4, 6, 2, 6, 1, 3, 1, 4, 5, 2, 3, 3, 4, 6, 3, 6, 3, 3, 3, 3, 5.
In order to calculate the experimental probability of each value occurring, tally the results.
1 2 3 4 5 6
13 12 21 19 16 19
Based on these results, the experimental probability of tossing a 1 is , or 0.13, and so forth. Now consider some important questions. Will the same values for these experimental probabilities occur if the same die is tossed another one hundred times? Is tossing a 3 on this die more likely than tossing a 2 because the experimental probability of tossing a 3 is , whereas the experimental probability of tossing a 2 is only ?
The answer to the first question is rather obvious. Although it is possible for the same experimental probabilities to recur, it is extremely unlikely.
The second question is more interesting and, in fact, not enough information is known. The difficulty arises in not knowing whether the die being tossed is fair or not.
Is a Die Fair?
A fair die is one for which the probability of tossing any of the six possible outcomes is equal, with each having a probability of , or approximately 0.167, of occurring. Dice manufacturers work hard to produce fair dice because games of chance are based on the premise that the six outcomes are equally likely. The experimental probabilities presented earlier were not exactly equal to , and so more investigation is necessary in order to determine if these particular die are fair.
Although there is no reason to believe the die being tossed is unfairly weighted on one side or another, this is certainly possible. If the die is not fair and the probability of tossing a 3 is somewhat greater than the probability of tossing the other values, this could be verified by tossing the die a much greater number of times. For example, if this die were tossed 10,000 times and we found the experimental probability of tossing a 3 to be around 0.21, we would be reasonably satisfied that the die is not fair. Conversely, if the experimental probability using 10,000 trials were about 0.16 or 0.17, we would assume the die is fair.
Assessing Experimental Results
In some circumstances it is impossible to calculate the theoretical probability that an event will occur. In these instances, data are gathered, an experimental probability is determined, and decisions are formed based on this information.
Consider the following scenario. A friend suggests playing a game of chance involving cutting a deck of cards. One person will be given the choice of calling "picture card" or "not picture card" before cutting the deck. If that person correctly guesses the category of the card selected, he or she wins the game, but if the guess is incorrect, he or she loses the game. After each game, the card is returned to the deck, and the deck is shuffled.
Given that there are only twelve picture cards in a standard deck of fifty-two cards, a smart strategy would be to guess "not picture card" every time. Using this strategy, the results of the first twenty games played are as follows: win, win, loss, win, win, loss, loss, loss, win, loss, win, win, loss, win, loss, win, win, loss, win, loss. The results of the first five games are not unexpected, but the loss of games 6, 7, and 8 may make the player feel unlucky. After completing twenty games, eleven were won and nine were lost. Although more games were won than lost, one may still be quite surprised by the results.
Because there are more than three times as many non-picture cards as picture cards, the player may have expected to win closer to fifteen or sixteen games rather than only eleven. Thus, the experimental probability of winning by guessing non-picture card based on this series of games is or 0.55. The theoretical probability, however, is , or approximately 0.77. The discrepancy between these values should lead the player to take a closer look at the deck of cards.
Upon inspecting the deck, the player finds that the friend had been using a pinochle deck, which is composed of forty-eight cards: eight nines, eight tens, eight jacks, eight queens, eight kings, and eight aces. Because twenty-four of these cards are picture cards (jacks, queens, and kings) and twenty-four are not picture cards (nines, tens, and aces), the probability of getting a non-picture card is also ½. The probability of getting a picture card is ½. Given this new information, the results attained seem more reasonable.
see also Games; Gaming; Probability and the Law of Large Numbers; Probability, Theoretical.
Robert J. Quinn
Holton, D. A. Probing Probability. New York: Holt, Rinehart and Winston, 1976.
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