Question

The probabilities associated with events A and B are shown in the Venn diagram.

The Venn diagram has 2 overlapping circles representing Event A and Event B. The non-overlapping section of Event A has 0.24, and of Event B has 0.36. The overlapping section of Event A and Event B has 0.12 and the section outside of Event A and Event B has 0.28.

What is the probability of event A if event B has already occurred?

A.


B.


C.


D.

Answer 1

To find the probability of event A given that event B has occurred, we use the formula for conditional probability:

\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]

From the information provided:

- The probability of event A occurring and not B (the non-overlapping section of A): \(P(A \cap B') = 0.24\)
- The probability of event B occurring and not A (the non-overlapping section of B): \(P(A' \cap B) = 0.36\)
- The probability of both events A and B occurring (the overlapping section): \(P(A \cap B) = 0.12\)
- The probability of neither event occurring (outside both A and B): \(P(A' \cap B') = 0.28\)

Now, we need to find \(P(B)\), which is the total probability of event B occurring. This can be calculated as follows:

\[
P(B) = P(A \cap B) + P(A' \cap B)
\]
Substituting the values we have:

\[
P(B) = 0.12 + 0.36 = 0.48
\]

Next, we can now calculate \(P(A | B)\):

\[
P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{0.12}{0.48}
\]

Calculating this gives:

\[
P(A | B) = \frac{12}{48} = \frac{1}{4} = 0.25
\]

Thus, the probability of event A given that event B has occurred is:

\[
\boxed{0.25}
\]