Question

Three chefs work in a kitchen. Based on predictions, the kitchen is idle 1 time out of 15; 2/15 of the time there is one order; 3 times out of 15 there are two orders; and 4/15 of the time there are three orders. Each order yields a net revenue of 10 dollars.

Let X be a random variable defined as the number of orders received in the kitchen.

a. Determine the probability distribution of X

b. Determine the cumulative distribution function of X

c. Calculate the probability that

i. All three chefs are working

ii. At least one of the chefs is working

d. Given that the owner of the kitchen pays 1 000 dollars in weekly fixed costs, what is the probability that he makes a profit if the profit function is given by the following relationship: P=1 200 X - 1 000 ?

e. What is the expected profit of the kitchen?

Answer 1

a) X : 0 1 2 3 4 or more

P(x) : 1/15 2/15 3/15 4/15 5/15

=0.0667 =0.1333 =0.2 =0.2667 =0.3333

(b) X : 0 1 2 3 4 or more

F(x): 1/15 3/15 6/15 10/15 15/15

=0.0667 =0.2 =0.4 =0.6667 1

( c ) (i) Probability that all the chefs are working

(ii) Probability that atleast one of the chefs is working

(d) P = 1200 X -1000

P takes positive value when X greater than equal to 1

the owner makes profit with probability 0.9333

(e)

Expected profit =E(P) = 1200.E(X) - 1000 = 2204