What is the probability that each person wins one match? (Hint: There are two different ways for this to happen.)?

P(A beats B) = 0.7, and thus P(B beats A) = 1 - 0.7 = 0.3
P(A beats C) = 0.6, and thus P(C beats A) = 1 - 0.6 = 0.4
P(B beats C) = 0.9, and thus P(C beats B) = 1 - 0.9 = 0.1

P(each person wins one match)
= P([(A beats B) and (B beats C) and (C beats A)] or [(B beats A) and (A beats C) and (C beats B)]
= P(A beats B) × P(B beats C) × P(C beats A) + P(B beats A) × P(A beats C) × P(C beats B)
= 0.7 × 0.9 × 0.4 + 0.3 × 0.6 × 0.1
= 0.27

This can happen if :
(a) A beats B but loses to C. Then B must beat C.
(b) A loses to B but beats C. Then C must beat B.

P(a) = P(A > B) x P(C > A) x P(B > C) = 0.7 x 0.4 x 0.9 = 0.252
P(b) = P(B > A) x P(A > C) x P(C > B) = 0.3 x 0.6 x 0.1 = 0.018.

So the probability of each person winning one match is 0.252 + 0.018 = 0.270.