Question

Let A and B be two independent events. The probability that both A and B happens is $\frac{1}{12}$ and the probability that neither A nor B happens is $\frac{1}{2}$. Then

• A. P(A)=$\frac{1}{3}$, P(B)=$\frac{1}{4}$
• B. P(A)=$\frac{1}{2}$, P(B)=$\frac{1}{6}$
• C. P(A)=$\frac{1}{6}$, P(B)=$\frac{1}{2}$
• D. P(A)=$\frac{2}{3}$, P(B)=$\frac{1}{8}$

Answer 1

Answer: (A)

P(A)=$\frac{1}{3}$, P(B)=$\frac{1}{4}$

Answer 2

The probability that both A and B happen is 1/12. This means P(A ∩ B) = 1/12.

The probability that neither A nor B happens is 1/2. This means the complement of the union of A and B (A∪B) doesn't occur. Since A and B are independent, we can express this probability as P(A') * P(B').

Now, let's use these probabilities to determine P(A) and P(B):

We know that P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Since A and B are independent, P(A ∩ B) = P(A) * P(B).

Given the probability that neither A nor B happens, we have: P(A' ∩ B') = 1/2. Since A and B are independent, this is P(A') * P(B').

Now, the probability of the complement of an event is simply 1 minus the probability of the event. So, P(A') = 1 - P(A), and P(B') = 1 - P(B).

Putting all this together, we can write:

1 - P(A) * (1 - P(B)) = 1/2

Solving for P(A), we get: P(A) = 1/3.

Similarly, we have:

1 - P(B) * (1 - P(A)) = 1/2

Solving for P(B), we get: P(B) = 1/4.

This justifies the answer: P(A) = 1/3 and P(B) = 1/4 based on the given probabilities of the events A and B.

The correct option is(A): P(A)=$\frac{1}{3}$, P(B)=$\frac{1}{4}$