Question

The probability of event A is x, and the probability of event B is y. If the two events are independent, which of these conditions must be true?

a. P(B|A) = y
b. P(A|B) = y
c. P(B|A) = x
d. P(A and B) = x + y

e. P(A and B) = x/y

P(A)

Answer 1

If events A and B are independent, the correct conditions are:

- P(B|A) = P(B)

- P(A ∩ B) = P(A) * P(B) = x * y.

These reflect that the probability of B given A is equal to the probability of B, and the probability of the intersection of A and B is the product of their individual probabilities.

Option D and C is correct.

If events A and B are independent, it means that the occurrence or non-occurrence of one event does not affect the probability of the other event. In mathematical terms, for independent events A and B:

P(A ∩ B) = P(A) * P(B)

This formula defines the probability of the intersection of events A and B (the occurrence of both events).

Now, let's analyze the provided options:

a. P(B|A) = P(B) = y

This is incorrect because the probability of B given A is not necessarily equal to the probability of B.

b. P(A|B) = P(A) = x

This is incorrect for the same reason as option (a). The probability of A given B is not necessarily equal to the probability of A.

c. P(B|A) = P(B) = y

This is correct. For independent events, the probability of B given A is equal to the probability of B.

d. P(A ∩ B) = P(A) * P(B) = x * y

This is correct. The probability of the intersection of A and B is the product of their individual probabilities.

e. P(A ∩ B) = x/y

This is incorrect. The correct expression for P(A ∩ B) is x * y.

In conclusion, the correct conditions for independent events A and B are options (c) and (d).

Answer 2

The right answer for the question that is being asked and shown above is that: "d. P(A and B) = x + y." The probability of event A is x, and the probability of event B is y. If the two events are independent, the condition must be true is this d. P(A and B) = x + y