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Transcript of 1 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison...
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
P(x) = probability of getting exactly x success among n trials
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 )
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
P(x) = • px • qn-x (n - x )! x!
Binomial Probability Formula
n !
Method 1
P(x) = nCx • px • qn-x
for calculators with nCr function, r = x
5Chapter 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
Method 1 – Using a formula
6Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
Method 1 – Using a formula
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 • 0.1 = 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
Method 1 – Using a formula
8Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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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 0123456789
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0.2060.3430.2670.1290.0430.0100.0020.0000.0000.0000.0000.0000.0000.0000.0000.000
P(x) x
Table A-1 Binomial Probability Distribution
For n = 15 and p = 0.10Method 2
9Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Example: Using Table A-1 for n = 15 and p = 0.10, find the following:
a) The probability of exactly 3 successesb) The probability of at most 3 successes
a) P(3) = 0.129
b) P(at most 3) = P(0 or 1 or 2 or 3)
= P(2) or P(1) or P(2) or P(3)
= 0.206 + 0.343 + 0.267 + 0.129
= 0.945
Note = This method is limited because a table may not be available for every n and/or p.
Method 2 – Using a table
10Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Probabilities with “Exact” successes
• Press 2nd, VARS (DISTR). • Select the option binompdf(). • Complete the entry binompdf(n, p, x)
to obtain P(x). – n is the number of trials– p is the probability of success–x is the EXACT number of
successes.
Method 3 – Using TI-83/4
11Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
• Example: What is the probability of getting exactly 2 heads when 4 tosses are made?
• Solution: –P(2) = binompdf(4, 0.5, 2)–P(2) = 0.375
Method 3 - Using TI-83/4
Probabilities with “Exact” successes
12Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Probabilities with “At most” successes
• Example: What is the probability of getting at most 2 heads when 4 tosses are made?
• Express at most 2 as an inequality.– P( x ≤ 2) which means x = 0 or 1 or 2
• Solution: – P( x ≤ 2) = P(0) + P(1) + P(2)– P( x ≤ 2) = 0.0625 + 0.25 + 0.375 = 0.6875– Where the probabilities would computed using binompdf(4,0.5, 0) then binompdf(4,0.5, 1) etc…
Method 3 - Using TI-83/4
13Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Probabilities with “At most” successes
• Press 2nd, VARS, select the option binomcdf().
• Note: The “c” indicates this is a cumulative function. It adds all the probabilities from zero up to x number of successes.
• Complete the entry to obtain P(At most x) = binomcdf(n, p, x), where x is the MAXIMUM number of successes.
Method 3 - Using TI-83/4
14Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
• Example: What is the probability of getting at most 2 heads when 4 tosses are made?
• Solution: – P( x ≤ 2) = binomcdf(4, 0.5, 2) = 0.6875.
Method 3 - Using TI-83/4
15Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Probabilities with “At least” successes
• When doing at least problems we must use the complement rule P(A) = 1 – P(not A)
• Complete the entry P(At least x) = 1 - binomcdf(n, p, x- 1),
where x is the MINIMUM number of successes.
Method 3 - Using TI-83/4
16Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
• Example: What is the probability of getting at least 3 heads when 4 tosses are made?
• Solution:– P(x≥3) = 1 – P(x ≤ 2)– P(x≥3) = 1 - binomcdf(4, 0.5, 2) = 0.3125.
Note: This is the same as • P( x ≥ 3)= P(x=3)+ P(x=4)• P( x ≥ 3)= 0.25 + 0.0625 = 0.3125
Method 3 - Using TI-83/4
17Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Recap
• P(x) Ask to find the probability of EXACT number of successes.
– Formula: P(x) = nCx· px · qn-x
– Calculator: P(x) = binompdf(n,p,x)
• P(X x) Ask to find the probability of AT MOST a number of successes.
– Calculator: P(X x ) = binomcdf(n, p, x)
• P(X x) Ask to find the probability of AT LEAST a number of successes.
– Calculator: P(X x ) = 1 - binomcdf(n, p, x-1)