Question

Let A and B be two events such that P(A) = 0.27, P(B) = 0.35 and P(A Or B) = 0.38. Calculate P(A|B). Give your answer as a decimal to the nearest tenth.

Answer 1

To find \( P(A|B) \), we can use the formula:

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

First, we need to find \( P(A \cap B) \). We can use the inclusion-exclusion principle for probabilities, which states:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

We know:

\[
P(A) = 0.27, \quad P(B) = 0.35, \quad P(A \cup B) = 0.38
\]

Plugging the known values into the formula gives:

\[
0.38 = 0.27 + 0.35 - P(A \cap B)
\]

Now, let's combine the values to solve for \( P(A \cap B) \):

\[
0.38 = 0.62 - P(A \cap B)
\]

Rearranging gives:

\[
P(A \cap B) = 0.62 - 0.38 = 0.24
\]

Now that we have \( P(A \cap B) \), we can calculate \( P(A|B) \):

\[
P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.24}{0.35}
\]

Calculating the fraction yields:

\[
P(A|B) = \frac{0.24}{0.35} \approx 0.6857
\]

Rounding to the nearest tenth, we get:

\[
P(A|B) \approx 0.7
\]

Thus, the final answer is:

\[
\boxed{0.7}
\]