1. Copyright © Cengage Learning. All rights reserved. The Binomial Probability Distribution and Related Topics 6

2. Copyright © Cengage Learning. All rights reserved. Section 6.3 Additional Properties of the Binomial Distribution

3. 3 Focus Points Make histograms for binomial distributions. Compute  and  for a binomial distribution. Compute the minimum number of trials n needed to achieve a given probability of success P (r).

4. 4 Graphing a Binomial Distribution

5. 5 Any probability distribution may be represented in graphic form. How should we graph the binomial distribution? Remember, the binomial distribution tells us the probability of r successes out of n trials. Therefore, we’ll place values of r along the horizontal axis and values of P (r) on the vertical axis. The binomial distribution is a discrete probability distribution because r can assume only whole-number values such as 0, 1, 2, 3,... Therefore, a histogram is an appropriate graph of a binomial distribution.

6. 6 Graphing a Binomial Distribution Procedure:

7. 7 Example 6 – Graph of a Binomial Distribution A waiter at the Green Spot Restaurant has learned from long experience that the probability that a lone diner will leave a tip is only 0.7. During one lunch hour, the waiter serves six people who are dining by themselves. Make a graph of the binomial probability distribution that shows the probabilities that 0, 1, 2, 3, 4, 5, or all 6 lone diners leave tips.

8. 8 Example 6 – Solution This is a binomial experiment with n = 6 trials. Success is achieved when the lone diner leaves a tip, so the probability of success is 0.7 and that of failure is 0.3: n = 6 p = 0.7 q = 0.3

9. 9 Example 6 – Solution We want to make a histogram showing the probability of r successes when r = 0, 1, 2, 3, 4, 5, or 6. It is easier to make the histogram if we first make a table of r values and the corresponding P (r) values (Table 6-12). cont’d Binomial Distribution for n = 6 and p = 0.70 Table 6-12

10. 10 Example 6 – Solution We’ll use Table 2 of the Appendix to find the P(r) values for n = 6 and p = 0.70. To construct the histogram, we’ll put r values on the horizontal axis and P(r) values on the vertical axis. Our bars will be one unit wide and will be centered over the appropriate r value. cont’d

11. 11 Example 6 – Solution The height of the bar over a particular r value tells the probability of that r (see Figure 6-3). cont’d Graph of the Binomial Distribution for n = 6 and p = 0.7 Figure 6-3

12. 12 Example 6 – Solution The probability of a particular value of r is given not only by the height of the bar over that r value but also by the area of the bar. Each bar is only one unit wide, so its area (area = height times width) equals its height. Since the area of each bar represents the probability of the r value under it, the sum of the areas of the bars must be 1. In this example, the sum turns out to be 1.001. It is not exactly equal to 1 because of rounding error. cont’d

13. 13 Mean and Standard Deviation of a Binomial Distribution

14. 14 Mean and Standard Deviation of a Binomial Distribution Two other features that help describe the graph of any distribution are the balance point of the distribution and the spread of the distribution about that balance point. The balance point is the mean  of the distribution, and the measure of spread that is most commonly used is the standard deviation . The mean  is the expected value of the number of successes.

15. 15 Mean and Standard Deviation of a Binomial Distribution For the binomial distribution, we can use two special formulas to compute the mean  and the standard deviation .

16. 16 Mean and Standard Deviation of a Binomial Distribution Procedure:

17. 17 Example 7 – Compute  and  Let’s compute the mean and standard deviation for the distribution of Example 6 that describes that probabilities of lone diners leaving tips at the Green Spot Restaurant. Solution: In Example 6, n = 6 p = 0.7 q = 0.3 For the binomial distribution,  = np = 6(0.7) = 4.2

18. 18 The balance point of the distribution is at  = 4.2. The standard deviation is given by cont’d Example 7 – Solution

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