1. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 12.11 Binomial Probability Formula

2. Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Binomial Probability Formula 12.11-2

3. Copyright 2013, 2010, 2007, Pearson, Education, Inc. To Use the Binomial Probability Formula There are n repeated independent trials. Each trial has two possible outcomes, success and failure. For each trial, the probability of success (and failure) remains the same. 12.11-3

4. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Binomial Probability Formula The probability of obtaining exactly x successes, P(x), in n independent trials is given by: where p is the probability of success on a single trial and q (= 1 – p) is the probability of failure on a single trial. 12.11-4

5. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Colored Balls with Replacement A basket contains 3 balls: 1 red, 1 blue, and 1 yellow. Three balls are going to be selected with replacement from the basket. Find the probability that a.no red balls are selected. 12.11-5

6. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Colored Balls with Replacement Solution p = 1/3, q = 1 – 1/3 = 2/3 12.11-6

7. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Colored Balls with Replacement b. exactly 1 red ball is selected. Solution p = 1/3, q = 1 – 1/3 = 2/3 12.11-7

8. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Colored Balls with Replacement c. exactly 2 red balls are selected. Solution p = 1/3, q = 1 – 1/3 = 2/3 12.11-8

9. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Colored Balls with Replacement d. exactly 3 red balls are selected. Solution p = 1/3, q = 1 – 1/3 = 2/3 12.11-9

10. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Quality Control for Batteries A manufacturer of batteries knows that 0.4% of the batteries produced by the company are defective. a) Write the binomial probability formula that would be used to determine the probability that exactly x out of n batteries produced are defective. 12.11-10

11. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Quality Control for Batteries Solution p = 0.4% = 0.004 q = 1 – 0.004 = 0.996 12.11-11

12. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Quality Control for Batteries b) Write the binomial probability formula that would be used to find the probability that exactly 3 batteries of 75 produced will be defective. Do not evaluate. 12.11-12

13. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Quality Control for Batteries Solution x = 3 n = 75 12.11-13

14. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Planting Trees The probability that a tree planted by a landscaping company will survive is 0.8. Determine the probability that a) none of four trees planted will survive. b) at least one of four trees planted will survive. 12.11-14

15. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Planting Trees Solution a) p = 0.8, q = 0.2, x = 0, n = 4 12.11-15

16. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Planting Trees Solution b) Probability that at least 1 tree survives can be found by subtracting the probability that none survives from 1. 12.11-16