Question

A restaurant needs a staff of 3 waiters and 2 chefs to be properly staffed. The joint probability model for the number of waiters (X) and chefs (Y) that show up on any given day is given below.

	X
Y		0	1	2	3
	0	k	0.01	0.01	0.02
	1	0.02	0.04	0.06	0.04
	2	0.01	0.03	0.05	0.65

a) What must the value of k be for this to be a valid probability model?  

b) What is the probability that at least one waiter and at least one chef show up on any given day?  

c) What is the probability that more chefs show up than waiters on any given day?  

d) What is the probability that more than three total staff (waiters and chefs) will show up on any given day?  

e) What is the expected total number of staff (waiters and chefs) that will show up on any given day?  

f) What is the probability that three waiters will show up on any given day?  

g) What is the probability that two chefs will show up on any given day?  

X and Y are independent.

TrueFalse    

Answer 1

a)

as sum of probability is 1, therefore k=0.06

b) probability that at least one waiter and at least one chef show up on any given day=1-0.06-0.01-0.02=0.91

c)probability that more chefs show up than waiters on any given day =0.06

d)

probability that more than three total staff =0.74

e)

expected total number of staff =4.09

f) probability that three waiters will show up on any given day =0.71

g) probability that two chefs will show up on any given day =0.74

X and Y are independent :False