Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 12.11 Binomial Probability Formula
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 12.11
Binomial Probability
Formula
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Binomial Probability Formula
12.11-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.11-3
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.
P x n
Cx pxqn x
12.11-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.11-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Selecting Colored Balls with Replacementc. exactly 2 red balls are
selected.Solutionp = 1/3, q = 1 – 1/3 = 2/3
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Selecting Colored Balls with Replacementd. exactly 3 red balls are
selected.Solutionp = 1/3, q = 1 – 1/3 = 2/3
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.11-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
12.11-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.11-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
12.11-13
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 thata) none of four trees planted will survive.b) at least one of four trees planted will survive.
12.11-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
12.11-15
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
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