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