Three friends (A, B, and C) will participate in a round-robin tournament in whic
ID: 3379679 • Letter: T
Question
Three friends (A, B, and C) will participate in a round-robin tournament in which each one plays both of the others. Suppose that
P(A beats B) = 0.5
P(A beats C) = 0.6
P(B beats C) = 0.9
and that the outcomes of the three matches are independent of one another.
(a) What is the probability that A wins both her matches and that B beats C?
(b) What is the probability that A wins both her matches?
(c) What is the probability that A loses both her matches?
(d) What is the probability that each person wins one match? (Hint: There are two different ways for this to happen.)
Explanation / Answer
P(A beats B) = 0.5
therefore P(B beats A)=1-0.5=0.5
P(A beats C) = 0.6
therefore P(C beats A)=1-0.6=0.4
P(B beats C) = 0.9
therefore P(C beats B)=1-0.9=0.1
(a) P(A wins both matches and B beats C)
= P(A beatsB)*P(A beats C)*P(B beats C)
= 0.5*0.6*0.9
= 0.27
P(A wins both matches and B beats C) = 0.27
(b) P(A wins both matches)
= P(A beatsB)*P(A beats C)
= 0.5*0.6
= 0.3
P(A wins both matches) = 0.3.
(c) P(A loses both matches)
= P(B beatsA)*P(C beatsA)
= 0.5*0.4
= 0.2
P(A loses both matches)=0.2
(d) P(each person wins one match)
= P(A beats B)*P(B beats C)*P(C beats A)+ P(A beats C)*P(C beats B)* P(B beats A)
= 0.5*0.9*0.4+0.6*0.1*0.5
= 0.21
P(each person wins one match) = 0.21