Question

A used car salesperson claims that the probability of he selling a used car to an individual looking to purchase a used car is 70% and this probability does not vary from individual to individual. Suppose 5 individuals come to speak to this salesperson one day. If his belief is correct,

1. The probability (to 4 decimal places) that he will sell a car to the first individual he speaks to is ______

2. The probability (to 4 decimal places) that he will not sell a car to any of these 5 individuals is ______

3. The probability (to 4 decimal places) that he will sell a car to exactly 2 individuals is _______

4. The expected number (to 2 decimal places) of individuals he will sell a car to is ­­________

5. The standard deviation (to 4 decimal places) in the number of individuals to whom he will sell a car is

________

6. If the salesperson pays the dealership who allows him to sell cars $1000 per day for the privilege of working at the dealership, and, if he earns $2000 for each car he sells, the expected income per day is

________

Answer 1

probability of a used sales person selling a used car to an individual looking to purchase a used car :p=  70/100=0.7

probability of a used sales person not selling a used car to an individual looking to purchase a used car q:1-p=1-0.7 =0.3

Number of individual came to speak to this sales person on one day : n= 5

a. The probability that he will sell a car to the first individual he speaks to is = p =0.70

b. The probability that he will not sell a car to any of these 5 individuals =

Probability that he will not sell a car to first individual x Probability that he will not sell a car to second individual x Probability that he will not sell a car to third individual x Probability that he will not sell a car to fourth individual x Probability that he will not sell a car to fifth individual

= 0.3 x 0.3 x 0.3 x 0.3 x 0.3 =0.00243

The probability that he will not sell a car to any of these 5 individuals = 0.0024

c.

X : Number of individuals he will sell car to

X follows Binomial distribution with n=5 and p=0.7

Probability that he will sell a car to r individuals is = P(X=r)

The probability that he will sell a car to exactly 2 individuals is

The probability that he will sell a car to exactly 2 individuals is 0.1323

Mean of binomial distribution =np

Standard deviation of binomial distribution =

c.

The expected number of individuals he will sell a car to is = np = 5 x 0.7 =3.5

The expected number of individuals he will sell a car to is 3.5

d. The standard deviation (to 4 decimal places) in the number of individuals to whom he will sell a car is

The standard deviation (to 4 decimal places) in the number of individuals to whom he will sell a car is 1.0247

e.

If the salesperson pays the dealership who allows him to sell cars $1000 per day for the privilege of working at the dealership, and, if he earns $2000 for each car he sells, the expected income per day is

Amount paid by the salesperson to the dealership per day = $1000

Amount earned by the sales person for each car he sells = $2000

Expected amount earned by the sales =2000 x  The expected number of individuals he will sell a car to = 2000 x 3.5 = 7000

expected income per day

= Expected amount earned by the sales - Amount paid by the salesperson to the dealership per day

= 7000 - 1000 =6000

expected income per day is $6000