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.)

Answer 1

The student is asked to find the probability of various outcomes in a round-robin tournament with three friends. Using the multiplication rule and the independence of events, we calculate each probability based on the given probabilities of one friend beating another in individual matches.

Step-by-step explanation:

The question involves understanding probabilities in a round-robin tournament and calculating several different probabilities based on given probabilities of one friend beating another. To solve these problems, we use the independence of events and the multiplication rule of probability.

• a. To find the probability that A wins both her matches and that B beats C, we multiply the probabilities of each independent event happening: P(A beats B) × P(A beats C) × P(B beats C) = 0.5 × 0.6 × 0.9 = 0.27.
• b. To find the probability that A wins both her matches, we multiply the probability of A beating B with the probability of A beating C: P(A beats B) × P(A beats C) = 0.5 × 0.6 = 0.3.
• c. To find the probability that A loses both her matches, we consider the complementary probabilities: (1 - P(A beats B)) × (1 - P(A beats C)) = (1 - 0.5) × (1 - 0.6) = 0.5 × 0.4 = 0.2.
• d. For each person to win one match, we can have two scenarios. First, A beats B, B beats C, and C beats A. The probability of this occurring is (0.5) × (0.9) × (1 - 0.6) = 0.18. Second, B beats A, C beats B, and A beats C. The probability of this occurring is (1 - 0.5) × (1 - 0.9) × (0.6) = 0.03. The total probability is the sum of both scenarios: 0.18 + 0.03 = 0.21.

Answer 2

0.27 ; 0.30 ; 0.20 ; 0.21

Step-by-step explanation:

Given that :

P(A beats B) = 0.5

P(A beats C) = 0.6

P(B beats C) = 0.9

Since the outcomes are independent :

A.) probability that A wins both her matches and that B beats C

P(A beats B) * P(A beats C) * P(B beats C)

0.5 * 0.6 * 0.9 = 0.27

B.) probability that A wins both her matches

P(A beats B) * P(A beats C)

0.5 * 0.6 = 0.3

C.) probability that A loses both her matches?

(1 - P(A beats B)) * (1 - P(A beats C)

(1 - 0.5) * (1 - 0.6)

0.5 * 0.4 = 0.20

D.) probability that each person wins one match

Either (A beats B), (B beats C). (C beats A) OR (A beats C), (C beats B), (B beats A)

Hence;

(0.5 * 0.9 * (1 - 0.6)) + (0.6 * (1 - 0.9) * (1 - 0.5)) = 0.21