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 <em>X</em> = number of sales made by the salesperson.
The probability that a potential customer makes a purchase is 0.10.
The salesperson contacts <em>n</em> = 8 potential customers per day.
The random variable <em>X</em> follows a Binomial distribution with parameters <em>n</em> and <em>p</em>.
The probability mass function of <em>X</em> is:
![P(X=x)={8\choose x}0.10^{x}(1-0.10)^{8-x};\ x=0,1,2,3...](https://tex.z-dn.net/?f=P%28X%3Dx%29%3D%7B8%5Cchoose%20x%7D0.10%5E%7Bx%7D%281-0.10%29%5E%7B8-x%7D%3B%5C%20x%3D0%2C1%2C2%2C3...)
(a)
Compute the probability the salesperson will make exactly two sales in a day as follows:
![P(X=2)={8\choose 2}0.10^{2}(1-0.10)^{8-2}\\=28\times 0.01\times 0.5314\\=0.1488](https://tex.z-dn.net/?f=P%28X%3D2%29%3D%7B8%5Cchoose%202%7D0.10%5E%7B2%7D%281-0.10%29%5E%7B8-2%7D%5C%5C%3D28%5Ctimes%200.01%5Ctimes%200.5314%5C%5C%3D0.1488)
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)
![=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](https://tex.z-dn.net/?f=%3D1-%7B8%5Cchoose%200%7D0.10%5E%7B0%7D%281-0.10%29%5E%7B8-0%7D-%7B8%5Cchoose%201%7D0.10%5E%7B1%7D%281-0.10%29%5E%7B8-1%7D%5C%5C%3D1-0.4305-0.3826%5C%5C%3D0.1869)
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:
![P(X=0)={8\choose 0}0.10^{0}(1-0.10)^{8-0}\\=8\times 1\times 0.4305\\=0.4305](https://tex.z-dn.net/?f=P%28X%3D0%29%3D%7B8%5Cchoose%200%7D0.10%5E%7B0%7D%281-0.10%29%5E%7B8-0%7D%5C%5C%3D8%5Ctimes%201%5Ctimes%200.4305%5C%5C%3D0.4305)
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:
![E(X)=np=8\times 0.10=0.80](https://tex.z-dn.net/?f=E%28X%29%3Dnp%3D8%5Ctimes%200.10%3D0.80)
Thus, the expected number of sales per day is 0.80.