three spies A B and C are in the running for an assignment i

three spies A, B, and C are in the running for an assignment in afghanistan. A is twice as likely to be selected as B, and C is three times as likely as a to be selected. what is the probability that A is selected?

Solution

We have that P(A) = 2P(B) and P(C) = 3P(A). By the axioms of probability, we also have that P(A) + P(B) + P(C) = 1.

From the latter equation, we get P(C) = 1 - P(A) - P(B). Substituting this into the first two equations gives us the system

P(A) = 2P(B)

1 - P(A) - P(B) = 3P(A)

From the first, P(B) = P(A) / 2. Substituting this into the second gives

1 - P(A) - P(A) / 2 = 3P(A)

1 - P(A) / 2 = 3P(A)

2 - P(A) = 6P(A)

2 = 7P(A)

so that

P(A) = 2/7 or about 28.6 %