This question is missing key numerical information, so the correct option cannot be determined as written. The standard form of this question is: “Given independent events A and B such that P(A)=0.3P(A)=0.3P(A)=0.3 and P(B)=0.5P(B)=0.5P(B)=0.5, which of the following is a correct statement?”
How independence is used
For independent events AAA and BBB, the joint probability satisfies P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B)P(A∩B)=P(A)P(B). Using the definition of conditional probability, P(A∣B)=P(A∩B)P(B)P(A\mid B)=\frac{P(A\cap B)}{P(B)}P(A∣B)=P(B)P(A∩B), so for independent events this becomes P(A∣B)=P(A)P(A\mid B)=P(A)P(A∣B)=P(A).
Applying to the usual numbers
With P(A)=0.3P(A)=0.3P(A)=0.3 and P(B)=0.5P(B)=0.5P(B)=0.5, independence gives P(A∣B)=0.3P(A\mid B)=0.3P(A∣B)=0.3. In the common multiple‑choice versions of this problem, the correct statement is therefore P(A∣B)=0.3P(A\mid B)=0.3P(A∣B)=0.3.
Typical option table
In a standard version, the options might look like:
Option statement| Correct?| Reason
---|---|---
P(A∣B)=0P(A\mid B)=0P(A∣B)=0| No| Independence does not force conditional
prob. 0. 4
P(B∣A)=0.3P(B\mid A)=0.3P(B∣A)=0.3| No| Should equal P(B)=0.5P(B)=0.5P(B)=0.5
if independent. 4
P(A∣B)=0.5P(A\mid B)=0.5P(A∣B)=0.5| No| Should equal P(A)=0.3P(A)=0.3P(A)=0.3
if independent. 4
P(A∣B)=0.3P(A\mid B)=0.3P(A∣B)=0.3| Yes| Equals P(A)P(A)P(A) for independent
events. 8
If your version has different numbers or answer choices, the same method applies: for independent events, the correct conditional statement always has P(A∣B)=P(A)P(A\mid B)=P(A)P(A∣B)=P(A) and P(B∣A)=P(B)P(B\mid A)=P(B)P(B∣A)=P(B).
