copyright 2013, 2010, 2007, pearson, education, inc. section 12.11 binomial probability formula

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 12.11

Binomial Probability

Formula


What You Will Learn

Binomial Probability Formula

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To Use the Binomial Probability FormulaThere 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.

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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.

P x n

Cx pxqn x

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Example 1: Selecting Colored Balls with ReplacementA 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.

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Example 1: Selecting Colored Balls with ReplacementSolutionp = 1/3, q = 1 – 1/3 = 2/3

11 2

3

3

P x n

Cx pxqn x

P 0 3

C0 1

3

02

3

3 0

8

27

12.11-6


Example 1: Selecting Colored Balls with Replacementb. exactly 1 red ball is

selected.Solutionp = 1/3, q = 1 – 1/3 = 2/3

3

1

3

2

3

2

P x n

Cx pxqn x

P 1 3

C1 1

3

12

3

31

4

912.11-7


Example 1: Selecting Colored Balls with Replacementc. exactly 2 red balls are


3

1

3

22

3

1

P x n

Cx pxqn x

P 2 3

C2 1

3

22

3

3 2

2

912.11-8


Example 1: Selecting Colored Balls with Replacementd. exactly 3 red balls are


1

1

3

32

3

0

P x n

Cx pxqn x

P 3 3

C3 1

3

32

3

3 3

1

2712.11-9


Example 2: Quality Control for BatteriesA 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.

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Example 2: Quality Control for BatteriesSolutionp = 0.4% = 0.004q = 1 – 0.004 = 0.996

P x n

Cx pxqn x

P x n

Cx 0.004 x 0.996 n x

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Example 2: Quality Control for Batteriesb) 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.

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Example 2: Quality Control for BatteriesSolutionx = 3n = 75

P 3 75

C3 0.004 3 0.996 75 3

P 3 75

C3 0.004 3 0.996 72

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Example 4: Planting Trees

The probability that a tree planted by a landscaping company will survive is 0.8. Determine the probability thata) none of four trees planted will survive.b) at least one of four trees planted will survive.

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Example 4: Planting TreesSolutiona) p = 0.8, q = 0.2, x = 0, n = 4

1 10.2 4 0.0016

P x n

Cx pxqn x

P 0 4

C0 0.8 0 0.2 4 0

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Example 4: Planting TreesSolutionb) Probability that at least 1 tree survives can be found by subtracting the probability that none survives from 1.

P

at least one

survives

1 P

none

survives

1 0.0016 0.998412.11-16

copyright 2013, 2010, 2007, pearson, education, inc. section 12.11 binomial probability formula

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