Question

Suppose P(A)=0.4 and P(B)=0.5. (a) If P(A∣B)=0.51, what is P(______)?

Now, fill in the blank with the appropriate question:

(a) What is the probability of event A given that event B has occurred?

(b) Are events A and B independent?

(c) Are events A and B mutually exclusive?

Answer 1

We calculate conditional probability using the formula P(A|B) = P(A AND B) / P(B). To test for independence, check if P(A AND B) is equal to P(A)P(B). For mutual exclusivity, check if P(A AND B) is zero.

Step-by-step explanation:

The student is asking about the concept of conditional probability, which is the probability of an event occurring given that another event has already occurred. Specifically, they are asking, "What is the probability of event A given that event B has occurred?" This is denoted as P(A|B).

For two events to be independent, the occurrence of one should not affect the probability of the occurrence of the other. We can determine this by checking if P(A AND B) equals P(A)P(B). If it does, the events are independent.

To be mutually exclusive events, the occurrence of one event precludes the occurrence of the other. This means that if one happens, the other cannot. We can determine this by examining if P(A AND B) equals 0; if so, the events are mutually exclusive.

Using the provided information:
- P(A) = 0.2
- P(B) = 0.3
- A and B are independent events
- Therefore, P(A AND B) = P(A)P(B) = (0.2)(0.3) = 0.06