Q: A company says that whether it increases its dividends depends on whether its earnings increase. From this we know:

A) P(earnings increase | dividend increase) is not equal to P(earnings increase).

B) P(both dividend increase and earnings increase) = P(dividend increase).

C) P(dividend increase | earnings increase) is not equal to P(earnings increase).

The correct answer is “A” but as per my understanding, “C” should be the answer. The conditional probability of A given that B has occurred. In the above question, the increase of dividend depends on earning increase hence dividend is “A” and earnings “B”. Where am I going wrong here in my logic?

Thanks very much.

Where did you get this question? Option C does seem to be correct too but from the question, it is asking what we can conclude from the statement in the question (relating to the non-independence of the two events).

For option A), P(earnings increase | dividend increase) is equals to P(earnings increase) only if the event of “earnings increase” (event A) and “dividend increase” (event B) are independent.

P(A | B) = \frac{P(A ~and B)}{P(B)} = \frac{P(A) \times P(B)}{P(B)} = P(A)

Since the question mentioned that event A is dependent on event B (i.e. not independent), therefore:

P(A | B) \neq P(A)

Thanks million for taking the time to explain the question. It was from Schweser Qbank.

Have a great day!!

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