Calculate the following probabilities (to 2 decimals) using the table of joint probabilities.

A1 A2
B1 .4 .3
B2 .2 .1
a. P(A1|B2) =

b. P(B2|A1) =

c. Did you expect the answers to Parts (a) and (b) to be reciprocals? That is, did you expect that P(A1|B2) = 1 / P(B2|A1)?

Why is this impossible (unless both probabilities are 1)?

It would help if you proofread your questions before you posted them.

What is the value if A1?

To calculate the probabilities, we can use the definition of conditional probability:

a. P(A1|B2): This represents the probability of event A1 happening given that event B2 has already occurred. To calculate this probability, we need to use the joint probabilities table.

P(A1|B2) = P(A1 and B2) / P(B2)

Looking at the table, we see that P(A1 and B2) = 0.1 and P(B2) = 0.1.

Therefore, P(A1|B2) = 0.1 / 0.1 = 1.

b. P(B2|A1): This represents the probability of event B2 happening given that event A1 has already occurred. Similarly, we can use the joint probabilities table.

P(B2|A1) = P(A1 and B2) / P(A1)

From the table, we find that P(A1 and B2) = 0.1 and P(A1) = 0.4.

Therefore, P(B2|A1) = 0.1 / 0.4 = 0.25.

c. No, we would not expect P(A1|B2) and P(B2|A1) to be reciprocals unless both probabilities equal 1. The reason for this is that conditional probabilities are not guaranteed to be reciprocal in general, unless all events have a probability of 1.

Reciprocity of conditional probabilities holds true only when we have two independent events, which means that the occurrence of one event does not affect the probability of the other. In this case, P(A1|B2) and P(B2|A1) are not independent events since their probabilities are not equal to 1. Therefore, we do not expect them to be reciprocals.