Transcript Ch 4.3 PowerPt

ELEMENTARY
Section 4-3
STATISTICS
Binomial Probability Distributions
EIGHTH
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION
MARIO F. TRIOLA
1
Definitions
Binomial 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.
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
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.
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 1
Binomial Probability
Formula
 P(x) =
n!
•
(n - x )! x!
 P(x) = nCx • px
px •
•
n-x
q
qn-x
for calculators with nCr function, r = x
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 1 – Using a formula
Binomial Probability Formula
P(x) =
n!
•
(n - x )! x!
Number of
outcomes with
exactly x
successes
among n trials
px •
n-x
q
Probability of x
successes
among n trials
for any one
particular order
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Method 1 – Using a formula
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.
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Method 1 – Using a formula
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.
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
3
2
• 0.9 • 0.1 = 0.0729
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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For n = 15 and p = 0.10
Method 2
Table A-1
Binomial Probability Distribution
n
x
P(x)
x
P(x)
15
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.206
0.343
0.267
0.129
0.043
0.010
0.002
0.0+
0.0+
0.0+
0.0+
0.0+
0.0+
0.0+
0.0+
0.0+
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.206
0.343
0.267
0.129
0.043
0.010
0.002
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 2 – Using a table
Example: Using Table A-1 for n = 15 and p = 0.10,
find the following:
a) The probability of exactly 3 successes
b) 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.
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 3 – Using TI-83/4
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.
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 3 - Using TI-83/4
Probabilities with “Exact” successes
• 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
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 3 - Using TI-83/4
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…
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Method 3 - Using TI-83/4
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.
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 3 - Using TI-83/4
• 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.
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 3 - Using TI-83/4
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.
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Method 3 - Using TI-83/4
• 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
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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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)
Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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