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 0 0.08 0.07 0.04 0.00 1 0.05 0.15 0.04 0.04 X1 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 0.06 (a) What is P(X₁ = 1, x2 = 1), that is, the probability that there is exactly one customer in each line? P(X₁ =1, X21) = (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. OA = {X₁ ≥ 2+ X₂ U X₂ ≤ 2 + X₁} = + 1 2 OA = {x₁ ≥2+ X₂ UX₂ = 2 + X₁ OA = {x₁ ≤ 2+ X₂UX ≤ 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) =
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 0 0.08 0.07 0.04 0.00 1 0.05 0.15 0.04 0.04 X1 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 0.06 (a) What is P(X₁ = 1, x2 = 1), that is, the probability that there is exactly one customer in each line? P(X₁ =1, X21) = (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. OA = {X₁ ≥ 2+ X₂ U X₂ ≤ 2 + X₁} = + 1 2 OA = {x₁ ≥2+ X₂ UX₂ = 2 + X₁ OA = {x₁ ≤ 2+ X₂UX ≤ 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) =
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Transcribed Image Text: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
0
0.08
0.07
0.04
0.00
1
0.05
0.15
0.04 0.04
X1
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
0.06
(a) What is P(X₁ = 1, x2 = 1), that is, the probability that there is exactly one customer in each line?
P(X₁ =1, X21) =
(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.
OA = {X₁ ≥ 2+ X₂ U X₂ ≤ 2 + X₁}
=
+
1
2
OA = {x₁ ≥2+ X₂ UX₂ = 2 + X₁
OA = {x₁ ≤ 2+ X₂UX ≤ 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) =
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