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(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,

Awhat is the probability that one or more customers will be turned away on a given day ? equation editorEquation Editor
Bwhat is the probability that the center's capacity will not be fully utilized on a day ? equation editorEquation Editor
Cby how much must the capacity be increased so the probability of turning a customer away is less than 0.10 ? equation editorEquation Editor
DFind the mean and standard deviation of X.

Use four decimals.

Verified Answer :

nobillionaireNobley

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