1105 11.7 Basics of Probability (b) “At most 4” can be written as “2 or 3 or 4.” (A sum of 1 is meaningless here.) The events represented by “2,” “3,” and “4” are mutually exclusive. P1at most 42 = P12 or 3 or 42 = P122 + P132 + P142 (1) The sample space for this experiment includes the 36 possible pairs of numbers shown in Figure 18. The pair 11, 12 is the only one with a sum of 2, so P122 = 1 36 . Also P132 = 2 36 because both 11, 22 and 12, 12 give a sum of 3. The pairs 11, 32, 12, 22, and 13, 12 have a sum of 4, so P142 = 3 36 . P1at most 42 = 1 36 + 2 36 + 3 36 Substitute into equation (1). = 6 36 Add fractions. = 1 6 Write in lowest terms. S Now Try Exercise 23(c). Summary of Properties of Probability For any events E and F, the following hold true. 1. 0 … P1E2 … 1 2. P1a certain event2 = 1 3. P1an impossible event2 = 0 4. P1E′2 = 1 - P1E2 5. P1E or F2 = P1E´F2 = P1E2 + P1F2 - P1E¨F2 Binomial Probability A probability experiment may consist of a repeated number of independent trials 1n2 with only two possible outcomes. Consider the example of tossing a coin 5 times and observing the number of tails. In this experiment there are n = 5 independent trials, or coin tosses, and there are two possible outcomes, head or tail, for each trial. It is common to consider “obtaining a tail” as a success because it is the outcome of interest, so “obtaining a head” would be considered a failure. If a probability experiment consists of n independent trials with two possible outcomes for each trial, and the probabilities remain constant for each trial, then it is a binomial experiment. Recall that the expression C1n, r2 is equivalent to the binomial coefficient An r B. Binomial Probability Let p represent the probability of a success, and let q = 1 - p represent the probability of a failure. In a binomial experiment, the probability of obtaining exactly r successes in n trials is found as follows. P1r successes in n trials2 = an r b p rq n−r
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