Assume that Pr[E]=1/2, Pr[F|E]=3/5, and Pr[E′∩F′]=3/10.

Question

Compute the following probabilities: (1) Pr[F], (2) Pr[G ∩ E]

Answer 1

Assume that Pr[E]=1/2, Pr[F|E]=3/5, and Pr[E′∩F′]=3/10.
Compute the following probabilities: (1) Pr[F], (2) Pr[G ∩ E]

First, I assume that G is a typographical error for F.
Now we can start. Do you know the defining identity for conditional probability?

P(F|E) = P(EF) / P(E)
We want to find P(EF), so multiply both sides by P(E)
P(E) * P(F|E) = P(EF)
Substitute the known numbers
1/2 * 3/5 = PEF
P(EF)= 3/10 ((Request (2)))

Since P(E) = P(EF) + P(EF’),
P(EF’) = P(E) - P(EF)
P(EF’) = 1/2 - 3/10 = 2/10

To discover P(F), let’s find p(F’)
Since P(F’) = P(EF’) + P(F’E’)
P(F’) = 2/10 + 3/10 = 1/2
Since P(F) = 1-P(F’)
P(F) = 1 - 1/2 = 1/2 ((Request (1) ))