Respuesta :
Answer:
(a) The probability the salesperson will make exactly two sales in a day is 0.1488.
(b) The probability the salesperson will make at least two sales in a day is 0.1869.
(c) The percentage of days the salesperson does not makes a sale is 43.05%.
(d) The expected number of sales per day is 0.80.
Step-by-step explanation:
Let X = number of sales made by the salesperson.
The probability that a potential customer makes a purchase is 0.10.
The salesperson contacts n = 8 potential customers per day.
The random variable X follows a Binomial distribution with parameters n and p.
The probability mass function of X is:
[tex]P(X=x)={8\choose x}0.10^{x}(1-0.10)^{8-x};\ x=0,1,2,3...[/tex]
(a)
Compute the probability the salesperson will make exactly two sales in a day as follows:
[tex]P(X=2)={8\choose 2}0.10^{2}(1-0.10)^{8-2}\\=28\times 0.01\times 0.5314\\=0.1488[/tex]
Thus, the probability the salesperson will make exactly two sales in a day is 0.1488.
(b)
Compute the probability the salesperson will make at least two sales in a day as follows:
P (X ≥ 2) = 1 - P (X < 2)
= 1 - P (X = 0) - P (X = 1)
[tex]=1-{8\choose 0}0.10^{0}(1-0.10)^{8-0}-{8\choose 1}0.10^{1}(1-0.10)^{8-1}\\=1-0.4305-0.3826\\=0.1869[/tex]
Thus, the probability the salesperson will make at least two sales in a day is 0.1869.
(c)
Compute the probability that a salesperson does not makes a sale is:
[tex]P(X=0)={8\choose 0}0.10^{0}(1-0.10)^{8-0}\\=8\times 1\times 0.4305\\=0.4305[/tex]
The percentage of days the salesperson does not makes a sale is,
0.4305 × 100 = 43.05%
Thus, the percentage of days the salesperson does not makes a sale is 43.05%.
(d)
Compute the expected number of sales per day as follows:
[tex]E(X)=np=8\times 0.10=0.80[/tex]
Thus, the expected number of sales per day is 0.80.
