Question

Suppose

P
(
A
)
=
0.29 and
P
(
B
)
=
0.48. If
P
(
A
∣
B
)
=
0.146, what is
P
(
A
∩
B
)
?

Answer 1

`P(\text{A}\cap \text{B}) = 0.07008`.

Explanation:

The vertical bar here reads "given."
`P(\text{A}|\text{B})=0.146` means that is the probability that A is true given that B is true is 0.146.

Consider the formula for conditional probabilities:

`\displaystyle P(\text{A}|\text{B}) = \frac{P(\text{A}\cap\text{B})}{P(\text{B})}`.

Multiply both sides by
`P(\text{B})`, the probability of B:

`\displaystyle P(\text{A}|\text{B})\cdot P(\text{B}) = \frac{P(\text{A}\cap\text{B})}{P(\text{B})} \cdot P(\text{B})`.

`\displaystyle P(\text{A}|\text{B})\cdot P(\text{B}) = {P(\text{A}\cap\text{B})}`.

Both
`P(\text{A}|\text{B})` and
`P(\text{B})` are given, and the question is asking for
`P(\text{A}\cap\text{B})`.

`P(\text{A}\cap\text{B}) = P(\text{A}|\text{B})\cdot P(\text{B}) = 0.146 * 0.48=0.07008`.