CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 4. Discrete Probability Distribution

p(x) =

(

n

x

)

px( 1 −p)n−x

Substituting we have:p(x) =

( 3

x

)

(. 25 )x( 1 −. 25 )^3 −x

a. Forx=2,

p(x) =

(

3

2

)

(. 25 )x( 1 −. 25 )^3 −^2

= ( 3 )(. 25 )^2 ( 1 − 25 )^1

= 0. 14

The probability is 14% that exactly two out of three randomly selected calls will last longer than 3.8 minutes.

b. Here,x=0. We use the binomial probability formula,

p(x= 0 ) =

(

3

0

)

(. 25 )^0 ( 1 −. 25 )^3 −^0

=

3!

0!( 3 − 0 )!

(. 25 )^0 (. 75 )^3

= 0. 422

The probability is 42.2% that none of the three randomly selected calls will last longer than 3.8 minutes.

Example:

A car dealer knows that from past experience he can make a sale to 20% of the customers that he interacts with.
What is the probability that, in five randomly selected interactions, he will make a sale to

a. Exactly three customers?

b. At most one customer?

c. At least one customer?

d. Determine the probability distribution for the number of sales.

Solution:

The success here is making a sale to the customer. The probability that the seller makes a sale to any customer is
p=20%= 0 .2. The number of trials isn=5. The binomial probability formula for our case is

p(x) =

(

5

x

)

(. 25 )x(. 8 )^5 −x

a. Here we want the probability of exactly 3 sales,x=3:

p(x) =

(

5

3

)

(. 2 )^3 (. 8 )^5 −x= 0. 051

This means that the probability that the sales person makes exactly three sales in five attempts is 5.1%.