Question

A car salesman is required to make three sales each day. Experience show that if he visits a customer, the customer will purchase a car with probability 0.2. (1) Calculate the probability that on a random day, the salesman makes exactly three unsuccessful visits (customer does not purahcse a car after the visit) before making three sales. (2) Calculate the probability that on a random day, the salesman makes at least five visits in total when he succesfully complete three sales. (3) Calculate the expected value of total number of visits the salesman need to make when he succesfully complete three sales.

Answer 1

1.0.008

2. 0.0307

3. 12

Explanation:

Probability p=0.2

Given that x + k = 3 and k = 3 hence x=0 where k is sales and x is successful visits

1

Required probability =P(x=0)

Using negative binomial distribution

=
`\binom{3-1}{3-1}0.2^30.8^(0)`

=
`\binom{2}{2}0.2^30.8^(0)=0.008`

2 ) Here , Given that , x + k = 5 and k = 3 hence x=5-3=2

Required probability =P(X=2)

Using negative binomial distribution

=
`\binom{5-1}{3-1}0.2^30.8^(2)=0.0307`

3 ) Given : k = 3 p = 0.2 q = 1 - p = 1 - 0.2 = 0.8

`E(X)=(kq)/(p)=(3*0.8)/(0.2)=12`