Let events A and B be such thatP (A ∪ B) = 34, P (A ∩ B) = 14 and P (Ac) = 23.Find P (B) and P (A ∩ Bc)
Question
Let events A and B be such thatP (A ∪ B) = 34, P (A ∩ B) = 14 and P (Ac) = 23.Find P (B) and P (A ∩ Bc)
Solution
Sure, I can help with that. Here's how you can find P(B) and P(A ∩ Bc):
Step 1: Find P(B)
We know that P(A ∪ B) = P(A) + P(B) - P(A ∩ B). We can rearrange this formula to solve for P(B):
P(B) = P(A ∪ B) + P(A ∩ B) - P(A)
We know that P(A ∪ B) = 3/4, P(A ∩ B) = 1/4, and P(A) = 1 - P(Ac) = 1 - 2/3 = 1/3. Substituting these values into the formula gives:
P(B) = 3/4 + 1/4 - 1/3 = 1 - 1/3 = 2/3
So, P(B) = 2/3.
Step 2: Find P(A ∩ Bc)
We know that P(A ∩ Bc) = P(A) - P(A ∩ B). We can substitute the known values into this formula:
P(A ∩ Bc) = P(A) - P(A ∩ B) = 1/3 - 1/4 = 1/12
So, P(A ∩ Bc) = 1/12.
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