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 cap 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: Recall the definition of conditional probability

Conditional probability \(P(A|B)\) is defined as \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) (\(P(B)> 0\)), which means the probability of event \(A\) occurring given that event \(B\) has occurred.

Step2: Analyze each option

• Option A: \(P(A)\) is the probability that event \(A\) will occur. It is not a conditional probability. Conditional probability has the form \(P(A|B)\) or \(P(B|A)\). So, option A is false.
• Option B: The probability that both event \(A\) and event \(B\) will occur is \(P(A\cap B)\), not \(P(A|B)\). So, option B is false.
• Option C: By the definition of conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) (\(P(B)>0\)), \(P(A|B)\) is the probability of event \(A\) occurring given that event \(B\) has occurred. So, option C is true.
• Option D: The probability that it is raining outside is \(P(\text{rain})\), which is an example of marginal (un - conditional) probability, not conditional probability. So, option D is false.
• Option E: The probability that it is both raining and cloudy is \(P(\text{rain}\cap\text{cloudy})\), which is an example of joint probability, not conditional probability. So, option E is false.
• Option F: The probability that it is raining if we already know it is cloudy outside is \(P(\text{rain}|\text{cloudy})\), which is an example of conditional probability. So, option F is true.

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.