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College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson 9 Probability and Statistics 9.3 Binomial Probability Probability Involving a Weighted Coin A coin is weighted so that the probability of heads is 0.6. • What is the probability of getting exactly two heads in five tosses of this coin? Probability Involving a Weighted Coin Since the tosses are independent, the probability of getting two heads followed by three tails is Probability Involving a Weighted Coin But this is not the only way we can get exactly two heads. • The two heads could occur, for example, on the second toss and the last toss. Probability Involving a Weighted Coin In this case, the probability is Probability Involving a Weighted Coin In fact, the two heads could occur on any two of the five tosses. • Thus, there are C(5, 2) ways in which this can happen, each with probability (0.6)2(0.4)3. • It follows that P(exactly 2 heads in 5 tosses) C(5,2) 0.6 0.4 2 0.023 3 Binomial Experiments The probabilities that we have just calculated are examples of binomial probabilities. • In general, a binomial experiment is one in which there are two outcomes, which we call “success” and “failure”. • In the coin-tossing experiment described previously, “success” is getting “heads” and “failure” is getting “tails”. • The following tells us how to calculate the probabilities associated with binomial experiments when we perform them many times. Binomial Probability An experiment has two possible outcomes. • S and F – called “success” and “failure” • With P(S) = p and P(F) = 1 – p. • The probability of getting exactly r successes in n independent trials of the experiment is P(r successes in n trials) = C(n, r)pr(1 – p)n – r Binomial Probability and Binomial Coefficient The name “binomial probability” is appropriate because C(n, r) is the same as the binomial n coefficient . r E.g. 1—Binomial Probability A fair die is rolled 10 times. Find the probability of each event. a) Exactly 2 sixes. b) At most 1 six. c) At least 2 sixes. E.g. 1—Binomial Probability We interpret “success” as getting a six and “failure” as not getting a six. • So, P(S) = 1/6 and P(F) = 5/6. • Since each roll of the die is independent from the others, we can use the formula for binomial probability with n = 10, p = 1/6. E.g. 1—Binomial Probability Example (a) Using these values in the formula gives us: P(exactly 2 are sixes) = C(10, 2)(1/6)2(5/6)8 ≈ 0.29 E.g. 1—Binomial Probability Example (b) The statement “at most 1 six” means 0 sixes or 1 six. So • So P(at most one six) = P(0 sixes or 1 six) = P(0 sixes) + P(1 six) = C(10, 0)(1/6)0(5/6)10 + C(10, 1)(1/6)1(5/6)9 ≈ 0.1615 + 0.3230 ≈ 0.4845 E.g. 1—Binomial Probability Example (c) The statement “at least two sixes” means two or more sixes. • Adding the probabilities that 2, 3, 4, 5, 6, 7, 8, 9, or 10 are sixes is a lot of work. • It’s easier to find the probability of the complement of this event. – The complement of “two or more are sixes” is “0 or 1 are sixes”. E.g. 1—Binomial Probability Example (c) So P(two or more sixes) = 1 – P(0 or 1 six) = 1 – 0.4845 = 0.5155 The Binomial Distribution The Binomial Distribution We can describe how the probabilities of an experiment are “distributed” among all the outcomes of an experiment by making a table of values. • The function that assigns to each outcome its corresponding probability is called a probability distribution. • A bar graph of a probability distribution in which the width of each bar is 1 is called a probability histogram. The next example illustrates these concepts. E.g. 4—A Binomial Distribution A fair coin is tossed eight times, and the number of heads is observed. • Make a table of the probability distribution, and draw a histogram. • What is the number of heads that is most likely to show up? E.g. 4—A Binomial Distribution This is a binomial experiment with n = 8 and p= ½, so 1 – p = ½ as well. • We need to calculate the probability of getting 0 heads, 1 head, 2 heads, 3 heads, and so on. E.g. 4—A Binomial Distribution To calculate the probability of 3 heads, we have 3 5 28 1 1 P(3 heads ) C(8,3) 256 2 2 E.g. 4—A Binomial Distribution The other entries in the following table are calculated similarly. E.g. 4—A Binomial Distribution We draw the histogram by making a bar for each outcome with width 1 and height equal to the corresponding probability. The Binomial Distribution Notice that the sum of the probabilities in a probability distribution is 1. • because the sum is the probability of the occurrence of any outcome in the sample space (this is the certain event).