If P(A) = 0.45, P(B) = 0.55, and P(A $\cup$ B) = 0.78, then P(A | B) =
A)zero
B)0.45
C)0.22
D)0.40

Question

Quizplus · Accepted Answer

We can use the formula for conditional probability: P(A|B) = P(A∩B)/P(B).
From the given information, we have:
P(A) = 0.45
P(B) = 0.55
P(A∩B) = P(A F1F1F1S1F1F1F10 B) = 0.78
We also know that:
A F1F1F1S1F1F1F10 B = (A∩F1F1F1S1)∪(F1F1F10 B)
So, we can apply the addition rule for probability to get:
P(A∩F1F1F1S1) + P(F1F1F10 B) - P(A∩F1F1F10 B) = 0.78
Since A and B are not mutually exclusive events, we need to use the general multiplication rule to calculate P(A∩F1F1F1S1):
P(A∩F1F1F1S1) = P(F1F1F1S1) P(A|F1F1F1S1)
We don't know P(F1F1F1S1) directly, but we can use the formula for total probability:
P(F1F1F1S1) = P(F1F1F1S1∩A) + P(F1F1F1S1∩A') = P(A∩F1F1F1S1) + P(A'∩F1F1F1S1)
We can simplify this as:
P(F1F1F1S1) = P(F1F1F1S1|A) P(A) + P(F1F1F1S1|A') P(A')
We don't know P(F1F1F1S1|A) or P(F1F1F1S1|A') directly either, but we can use the formula for conditional probability again to get:
P(F1F1F1S1|A) = P(F1F1F1S1∩A)/P(A)
We don't know P(F1F1F1S1∩A), but we can use the formula for intersection:
F1F1F1S1∩A = (A∩F1F1F1S1)⊆(F1F1F1S1F1F1F10 B)
So, we have:
P(F1F1F1S1|A) = P(A∩F1F1F1S1)/P(A) ≤ P(A F1F1F1S1F1F1F10 B)/P(A) = 0.78/0.45 < 2
Similarly:
P(F1F1F1S1|A') = P(F1F1F1S1∩A')/P(A') ≤ P(F1F1F10 B)/P(A') = (0.78 - P(A∩F1F1F10 B))/(1 - 0.45) = (0.78 - P(A∩F1F1F10 B))/0.55 < 2
We don't need the exact values of P(F1F1F1S1|A) and P(F1F1F1S1|A') to answer the question. We only need to know whether they are close to 0 or 1, because that would determine whether P(A∩F1F1F1S1) is close to 0.45*0.55 = 0.2475 or much smaller. If it is close to 0.2475, then P(A|F1F1F10 B) would be close to 0.4, the only answer choice that is close to 0.5*(0.78/0.55). If it is much smaller, then P(A|F1F1F10 B) would be closer to 0 or 0.22, the only answer choice that is less than 0.2475. We can also rule out 0.45 as an answer choice because it is impossible for P(A|B) to be equal to P(A) unless A and B are independent, which is not the case here. Therefore, the best choice is D, 0.4, because it is close to the expected value based on the given information.

If P(A) = 0 ∪\cup∪ B) = 078, Then P(A | B) = A)zero B)0

If P(A) = 0 $\cup$ B) = 078, Then P(A | B) =
A)zero
B)0