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THEORY OF PROBABILITY

The theory of probability develops mathematical representations of uncertainty. A fraction between zero and one, called the probability, describes the chance of each possible outcome or collection of outcomes. A set of possible outcomes is called an event. One interpretation of the probability fraction is that its numerator is the number of times we expect an event to happen; the denominator is the number of times the event could possibly happen. The probability associated with the combination of all possible outcomes must be one. The probability that a particular event will not happen is one minus the probability that it will happen. These principles can be clarified by a few simple illustrations.

For example, consider the probability associated with tossing a coin. The coin will fall either heads up or tails up when tossed. Two possible outcomes exist, and only one is the outcome of tossing a head. Therefore the probability of tossing a head is �. The probability that the coin will fall either heads or tails is �+� or one.

Another example is drawing a card at random from a deck of 52 playing cards. The chance that the ace of spades will be drawn from the deck is 1/52. Of the 52 cards, only one is the ace of spades. Of the 52 cards, four are aces. Thus the probability of drawing an ace of any suit is 4/52. The probability that a card drawn randomly will not be an ace is one minus the probability of drawing an ace, or 48/52.

Sometimes we need to know the probability of a particular combination of events. If the occurrence of one event does not change the probabilities associated with another event, the events are independent. We find the probability that two independent events will happen by multiplying together the separate probabilities that each event will happen. Consider the independent tossing of a nickel and a dime. The probability that both will fall heads up is � x �, or �, since the chance is � that each separate coin will fall heads up. The probability that at least one of the two coins will fall tails up is 3/4 (1 - 1/4). Confirm these results by looking at table 13-1 showing the different ways in which the coins may fall.

TABLE 13-1
Combinations of
Tossing Two Coins

Nickel

Dime

Heads up

Heads up

Heads up

Tails up

Tails up

Heads up

Tails up

Tails up

Only these four combinations can be made with the two coins. The first combination is the only one of the four that meets the condition that both coins land heads up. Conversely, the last three of the four meet the condition that at least one of the coins will fall tails up. The probability that three coins tossed simultaneously all fall heads up is found to be � x � x � or 1/8 in this same manner. This same pattern is followed for any number of coins tossed.

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