A certain market has both an express checkout line and a
superexpress checkout line.
Let X1 denote the number of customers
in line at the express checkout at a particular time of day, and
let X2 denote the number of customers
in line at the superexpress checkout at the same time. Suppose the
joint pmf
of X1 and X2 is
as given in the accompanying table.
x2
0
1
2
3
x1
0
0.09
0.06
0.04
0.00
1
0.05
0.15
0.05
0.04
2
0.05
0.04
0.10
0.06
3
0.00
0.04
0.04
0.07
4
0.00
0.02
0.05
0.05
(a) What is P(X1 =
1, X2 = 1), that is, the probability
that there is exactly one customer in each line?
P(X1 =
1, X2 = 1) =
(b) What
is P(X1 = X2),
that is, the probability that the numbers of customers in the two
lines are identical?
P(X1 = X2)
=
(c) Let A denote the event that there are at
least two more customers in one line than in the other line.
Express A in terms
of X1 and X2.
A =
{X1 ≤ 2
+ X2 ∪ X2 ≥ 2
+ X1}
A =
{X1 ≥ 2
+ X2 ∪ X2 ≤ 2
+ X1}
A =
{X1 ≥ 2
+ X2 ∪ X2 ≥ 2
+ X1}
A =
{X1 ≤ 2
+ X2 ∪ X2 ≤ 2
+ X1}
Calculate the probability of this event.
P(A) =
(d) What is the probability that the total number of customers in
the two lines is exactly four? At least four?
P(exactly four)
=
P(at least four)
=