If the probability is 0.54 that stock A will increase in value during the next month and the probability is 0.68 that stock B will increase in value during the next month, what is the greatest possible value for the probability that neither of these 2 events will occur?
A) 0.22
B) 0.32
C) 0.37
D) 0.46
E) 0.63

We want to make it AS HARD AS POSSIBLE for A and/or B to occur.
Strategy:
Make one of the probabilities DEPENDENT on the other.
In other words, make it so that one of the events can't happen UNLESS the other event happens.

Let's rephrase the problem so that one of the probabilities is more clearly dependent on the other.
Since the non-dependent event does NOT require the other event -- making it EASIER for the non-dependent event to happen -- the non-dependent event must have the GREATER of the two probabilities.

Let B = John buys a lottery ticket.
P(B) = 0.68.
Let A = John wins the lottery.
P(A) = 0.54.
Here, the probability of A is clearly dependent on the probability of B: John can win the lottery only if he first buys a ticket.
Question rephrased:

If the probability that John wins the lottery is 0.54, and the probability that John buys a lottery ticket is 0.68, what is the greatest possible 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

If John DOESN'T buy a lottery ticket, then NEITHER event (buying a ticket, winning the lottery) occurs.
P(John doesn't buy a lottery ticket) = 1 - 0.68 = 0.32.

The correct answer is B.

abhasjha wrote:

Answer: Option B

alternatively :

. Let's say
A = the event that stock A increases next month
B = the event that stock B increases next month
The prompt tell us
P(A) = 0.54
P(B) = 0.68
From here, we can calculate
P(not A) = 1 - 0.54 = 0.46
P(not B) = 1 - 0.68 = 0.32
We want the maximum overlap of P(not A) and P(not B), so even if they have full overlap, the size of that overlap region could only be as big as P(not B) = 0.32.