The events A and B are independent, and the probability that event A

Official Explanation

We're asked to find the probability that event B occurs, such that one of the answer choices reflects the probability of either A or B occurring—in other words, any outcome other than neither A nor B occurring.

Since the only outcome we want to exclude is neither A nor B occurring (see stimulus explanation, above), we can create an equation by subtracting that one undesired outcome from the total of 1:

Probability that at least one of A and B occurs = 1 – [0.6(1 – b)]

= 1 – (0.6 – 0.6b)
= 1 – 0.6 + 0.6b
= 0.4 + 0.6b

(Note that there are two other ways to set this equation up. We could add the three desired outcomes together to yield 0.4b + 0.4(1 – b) + 0.6b. Alternately, we could add the probabilities of A and B, then subtract the double-counted outcome of both occurring, yielding 0.4 + b – 0.4b. On Test Day, go with whichever approach feels most straightforward—all of these approaches simplify to 0.4 + 0.6b.)

Now let's consider the possible values given in the answer choice list. Thinking about the situation logically, we can determine that the probability that B occurs must be less than the probability that at least one of A or B occurs. We will therefore begin by plugging in the smaller values from our list for the variable b in the equation we set up above. If one of these values for b results in a probability for "at least one of the events A or B" that matches one of the given choices, we know we have found the correct answer.

If b were equal to 0.1, then the probability that at least one of A and B occurs would equal 0.46, which is not an answer choice. This strongly suggests that 0.25 is the probability that B occurs, since it is the only other small answer choice. A larger probability might be possible, since the probability of at least one of A and B might be as high as 0.8, but plugging in b = 0.25 is a reasonable next step.

We'll substitute b = 0.25 into our equation and see whether it produces one of the other answer choices. If it does, that pair will be correct. If it doesn't, we'll try a different value for b. Plugging b = 0.25 into our equation yields:

Probability that at least one of A and B occurs = 0.4 + 0.6(0.25)

= 0.4 + 0.15
= 0.55

This possible value matches one of the answer choices, so we have found the values that satisfy the conditions. Now, as always with Two-Part Analysis questions, we must be careful to select the correct answer choices in the appropriate columns; it would be a shame to do all the math correctly but lose the points for this question by mixing up the columns.

The correct response for the "probability that at least one of the events A and B occurs" is 0.55; the correct response for the "probability that event B occurs" is 0.25.