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The Binomial Probability Distribution and Related Topics 6 Copyright © Cengage Learning. All rights reserved. Section 6.2 Binomial Probabilities Copyright © Cengage Learning. All rights reserved. Binomial Experiment 3 Binomial Experiment 4 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula 5 Using a Binomial Distribution Table 6 Using a Binomial Distribution Table In many cases we will be interested in the probability of a range of successes. In such cases, we need to use the addition rule for mutually exclusive events. For instance, for n = 6 and p = 0.50, P(4 or fewer successes) = P(r 4) = P(r = 4 or 3 or 2 or 1 or 0) = P(4) + P(3) + P(2) + P(1) + P(0) 7 Using a Binomial Distribution Table It would be a bit of a chore to use the binomial distribution formula to compute all the required probabilities. Table 2 of Appendix gives values of P(r) for selected p values and values of n through 20. To use the table, find the appropriate section for n, and then use the entries in the columns headed by the p values and the rows headed by the r values. 8 Using a Binomial Distribution Table Table 6-10 is an excerpt from Table 2 of Appendix showing the section for n = 6. Notice that all possible r values between 0 and 6 are given as row headers. Excerpt from Table 2 of Appendix for n = 6 Table 6-10 The value p = 0.50 is one of the column headers. For n = 6 and p = 0.50, you can find the value of P(4) by looking at the entry in the row headed by 4 and the column headed by 0.50. Notice that P(4) = 0.234 (to three digits). 9 Using a Binomial Distribution Table Likewise, you can find other values of P(r) from the table. In fact, for n = 6 and p = 0.50, P(r 4) = P(4) + P(3) + P(2) + P(1) + P(0) = 0.234 + 0.312 + 0.234 + 0.094 + 0.016 = 0.890 Alternatively, to compute P(r 4) for n = 6 , you can use the fact that the total of all P(r) values for r between 0 and 6 is 1 and the complement rule. 10 Using a Binomial Distribution Table Since the complement of the event r 4 is the event r 5, we have P(r 4) = 1 – P(5) – P(6) = 1 – 0.094 – 0.016 = 0.890 11 Example 5 – Using the binomial distribution table to find P(r) A biologist is studying a new hybrid tomato. It is known that the seeds of this hybrid tomato have probability 0.70 of germinating. The biologist plants six seeds. a. What is the probability that exactly four seeds will germinate? Solution: This is a binomial experiment with n = 6 trials. Each seed planted represents an independent trial. We’ll say germination is success, so the probability for success on each trial is 0.70. 12 Example 5 – Solution n=6 p = 0.70 q = 0.30 cont’d r=4 We wish to find P(4), the probability of exactly four successes. In Table 2, Appendix, find the section with n = 6 (excerpt is given in Table 6-10). Excerpt from Table 2 of Appendix for n = 6 Table 6-10 13 Example 5 – Solution cont’d Then find the entry in the column headed by and the row headed by p = 0.70 and the row headed by r = 4. This entry is 0.324. P(4) = 0.324 14 Example 5 – Using the binomial distribution table to find P(r) cont’d b. What is the probability that at least four seeds will germinate? Solution: In this case, we are interested in the probability of four or more seeds germinating. This means we are to compute P(r 4). Since the events are mutually exclusive, we can use the addition rule P(r 4) = P(r = 4 or r = 5 or r = 6) = P(4) + P(5) + P(6) We already know the value of P(4). We need to find P(5) and P(6). 15 Example 5 – Solution cont’d Use the same part of the table but find the entries in the row headed by the r value 5 and then the r value 6. Be sure to use the column headed by the value of p, 0.70. P(5) = 0.303 and P(6) = 0.118 Now we have all the parts necessary to compute P(r 4). P(r 4) = P(4) + P(5) + P(6) = 0.324 + 0.303 + 0.118 = 0.745 16 Using Technology to Compute Binomial Probabilities 17 Using Technology to Compute Binomial Probabilities Some calculators and computer-software packages support the binomial distribution. In general, these technologies will provide both the probability P(r) for an exact number of successes r and the cumulative probability P(r k), where k is a specified value less than or equal to the number of trials n. Note that most of the technologies use the letter x instead of r for the random variable denoting the number of successes out of n trials. 18 Sampling Without Replacement: Use of the Hypergeometric Probability Distribution 19 Sampling Without Replacement: Use of the Hypergeometric Probability Distribution If the population is relatively small and we draw samples without replacement, the assumption of independent trials is not valid and we should not use the binomial distribution. The hypergeometric distribution is a probability distribution of a random variable that has two outcomes when sampling is done without replacement. This is the distribution that is appropriate when the sample size is so small that sampling without replacement results in trials that are not even approximately independent. A discussion of the hypergeometric distribution can be found in Appendix I of Understandable Statistics, 10th edition. 20