(1 point) based on recent records, the manager of a car painting center has determined the following probability distribution for the number of customers per day.
x 0 1 2 3 4 5
f(x) 0.04 0.18 0.27 0.19 0.13 0.19
if the center has the capacity to serve two customers per day,
(a) what is the probability that one or more customers will be turned away on a given day ? equation editorequation editor
(b) what is the probability that the center's capacity will not be fully utilized on a day ? equation editorequation editor
(c) by how much must the capacity be increased so the probability of turning a customer away is less than 0.10 ? equation editorequation editor
(d) find the mean and standard deviation of x.
use four decimals.

A.) The probability that one or more customers will be turned away implies the probability that 3 or more customers will visit the center. i.e. since the center has a capacilty of 2 customers, when more than 2 customers visit the center, they will turn the rest down.

P(turning down one or more customers) = P(3) + P(4) + P(5) = 0.19 + 0.13 + 0.19 = 0.51

b.) The probability that the center's capacity will not be utilized on a day implies the probability that one customer or no customer visits the center.

P(the center's capacity will not be fully utilized on a day) = P(0) + P(1) = 0.04 + 0.18 = 0.22

c.) For the probability of turning a customer away to be less than 0.10, the capacity of the center must be increased to five customers.

d.) Mean = E(x) = ∑xp(x) = 0(0.04) + 1(0.18) + 2(0.27) + 3(0.19) + 4(0.13) + 5(0.19) = 0 + 0.18 + 0.54 + 0.57 + 0.52 + 0.95 = 2.76

Standard deviation = sqrt(∑(x - E(x))^2 p(x)) = sqrt((0 - 2.76)^2 (0.04) + (1 - 2.76)^2 (0.18) + (2 - 2.76)^2 (0.27) + (3 - 2.76)^2 (0.19) + (4 - 2.76)^2 (0.13) + (5 - 2.76)^2 (0.19)) = sqrt(0.304704 + 0.557568 + 0.155952 + 0.010944 + 0.199888 + 0.953344) = sqrt(2.1824) = 1.4773