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(acap 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:

• Conditional probability $P(A|B)$ is the probability of event $A$ occurring given that event $B$ has occurred. So, statement C is correct.
• An example of conditional probability is the probability that it is raining given we already know it is cloudy outside. So, statement F is correct.
• Statement A is about simple probability of $A$, not conditional probability.
• Statement B is wrong as $P(A|B)$ is not the probability that both $A$ and $B$ occur.
• The probability that it is raining outside is a simple probability, not conditional (statement D).
• The probability that it is both raining and cloudy is a joint - probability, not conditional (statement E).

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.