Problem Solving

BTGmoderatorDC wrote: ↑

Sat Feb 01, 2020 11:55 pm

If the probability that Stock A will increase in value during the next month is 0.54, and the probability that Stock B will increase in value during the next month is 0.68. What is the greatest value for the probability that neither of these two events will occur?

A. 0.22
B. 0.32
C. 0.37
D. 0.46
E. 0.63



OA B

Source: GMAT Prep

Recall that if set U is the universal set containing two sets A and B, we have:

P(U) = P(A) + P(B) - P(A and B) + P(neither A nor B)

Since P(U) = 1, we can also say:

1 = P(A) + P(B) - P(A and B) + P(neither A nor B)

If we let the probability that stock A will increase be P(A) and the probability that stock B will increase be P(B), then we are given that P(A) = 0.54 and P(B) = 0.68. Thus, we can say:

1 = 0.54 + 0.68 - P(A and B) + P(neither A nor B)

Notice that we are being asked for the greatest value of P(neither A nor B). If that is the case, we want P(A and B) to be as large as possible since we are subtracting it. However, notice that P(A and B) can’t be larger than either P(A) or P(B). Therefore, P(A and B) can be only as large as the lesser value of P(A) and P(B). So, here P(A and B) can be as large as P(A), or 0.54. Thus:

1 = 0.54 + 0.68 - 0.54 + P(neither A nor B)

1 = 0.68 + P(neither A nor B)

0.32 = P(neither A nor B)

Answer: B