Question

which two statements about conditional probability are true?
a. conditional probability, ( p(a) ), is the probability that event ( a ) will occur.
b. conditional probability, ( p(a|b) ), is the probability that both event ( a ) and event ( b ) will occur.
c. conditional probability, ( p(a|b) ), is the probability of event ( a ) occurring given that event ( b ) has occurred.
d. the probability that it is raining outside is an example of conditional probability.
e. the probability that it is both raining and cloudy outside is an example of conditional probability.
f. the probability that it is raining if we already know it is cloudy outside is an example of conditional probability.

Explanation:

Step1: Analyze Option A

Conditional probability \(P(A)\) is not the probability that event \(A\) will occur. Conditional probability is about the probability of an event given another event has occurred. So, Option A is wrong.

Step2: Analyze Option B

Conditional probability \(P(A|B)\) is not the probability that both event \(A\) and event \(B\) will occur. The probability that both \(A\) and \(B\) occur is \(P(A\cap B)\). So, Option B is wrong.

Step3: Analyze Option C

By the definition of conditional probability, \(P(A|B)\) is the probability of event \(A\) occurring given that event \(B\) has occurred. So, Option C is correct.

Step4: Analyze Option D

The probability that it is raining outside is a simple probability (un - conditional), not a conditional probability. So, Option D is wrong.

Step5: Analyze Option E

The probability that it is both raining and cloudy outside is \(P(\text{rain}\cap\text{cloudy})\), which is a joint probability, not a conditional probability. So, Option E is wrong.

Step6: Analyze Option F

The probability that it is raining if we already know it is cloudy outside is a conditional probability. Let \(A\) be the event of "raining" and \(B\) be the event of "cloudy". Then it is \(P(A|B)\). So, Option F is correct.

Answer:

C. Conditional probability, \(P(A|B)\), is the probability of event \(A\) occurring given that event \(B\) has occurred.
F. The probability that it is raining if we already know it is cloudy outside is an example of conditional probability.