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S TATISTICS. E LEMENTARY. Section 4-3 Binomial Probability Distributions. M ARIO F . T RIOLA. E IGHTH. E DITION. Binomial Probability Distribution 1.The experiment must have a fixed number of trials . - PowerPoint PPT PresentationTRANSCRIPT
1Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH
EDITIONEDITION
ELEMENTARY STATISTICSSection 4-3 Binomial Probability Distributions
2Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
DefinitionsBinomial Probability Distribution
1. The experiment must have a fixed number of trials.
2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.)
3. Each trial must have all outcomes classified into two categories.
4. The probabilities must remain constant for each trial.
3Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Notation for Binomial Probability Distributions
n = fixed number of trials
x = specific number of successes in n trials
p = probability of success in one of n trials
q = probability of failure in one of n trials (q = 1 - p )
P(x) = probability of getting exactly x
success among n trials
Be sure that x and p both refer to the same category being called a success.
4Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
(n - x )! x! P(x) = • px • qn-x
Binomial Probability Formula
n !
Method 1
5Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
P(x) = • px • qn-x (n - x )! x!
Binomial Probability Formula
n !
Method 1
P(x) = nCx • px • qn-x
for calculators with nCr key, where r = x
6Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
This is a binomial experiment where:
n = 5
x = 3
p = 0.90
q = 0.10
Example: Find the probability of getting exactly 3 correct responses among 5 different requests from AT&T directory assistance. Assume in general, AT&T is correct 90% of the time.
7Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
This is a binomial experiment where:
n = 5
x = 3
p = 0.90
q = 0.10
Using the binomial probability formula to solve:
P(3) = 5C3 • 0.9 • 01 = 0.0.0729
Example: Find the probability of getting exactly 3 correct responses among 5 different requests from AT&T directory assistance. Assume in general, AT&T is correct 90% of the time.
3 2
8Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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P(x) n x
15
Table A-1
For n = 15 and p = 0.10
9Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
0123456789
101112131415
0.2060.3430.2670.1290.0430.0100.0020.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+
P(x) n x
15
Table A-1
For n = 15 and p = 0.10
10Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
0123456789
101112131415
0.2060.3430.2670.1290.0430.0100.0020.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+
P(x) n x
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P(x) x
Table A-1 Binomial Probability Distribution
For n = 15 and p = 0.10
11Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Example: Using Table A-1 for n = 5 and p = 0.90, find the following:
a) The probability of exactly 3 successesb) The probability of at least 3 successes
a) P(3) = 0.073
b) P(at least 3) = P(3 or 4 or 5)
= P(3) or P(4) or P(5)
= 0.073 + 0.328 + 0.590
= 0.991
12Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
P(x) = • px • qn-xn ! (n - x )! x!
Number of outcomes with
exactly x successes
among n trials
Binomial Probability Formula
13Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
P(x) = • px • qn-xn ! (n - x )! x!
Number of outcomes with
exactly x successes
among n trials
Probability of x successes
among n trials for any one
particular order
Binomial Probability Formula