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

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 0123456789

101112131415

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)