Two people are to be selected at random from a certain group that includes Claire and Max. What is the probability that the 2 people selected will include Claire but not Max?
(1) The probability that the 2 people selected will be Claire and Max is \(\frac{1}{15}\).This question seems tough, but we can evaluate this statement without using any math.
Notice that there will only be one case in which this statement will be true.
If, for instance, there were just 3 people in the group, the probability that the 2 people selected will be Claire and Max would be greater than \(\frac{1}{15}\).
Similarly, we can see without performing any exact calculations that, if there were 200 people in the group, the probability that the 2 people selected will be Claire and Max would be much less than \(\frac{1}{15}\).
So, in general, as the group gets larger, the probability that Claire and Max will be the two people chosen gets smaller. It has to because, the more people there are in the group, the greater the probability that someone other than them will will chosen.
Similarly, as the group gets smaller, the probability that Claire and Max will be the two people chosen must get smaller.
There is no way around that dynamic.
In other words, it's impossible for the probability that Claire and Max will be chosen to be the same for different group sizes.
Thus, only in one particular case will the probability be \(\frac{1}{15}\), and knowing that, we could work from that information to the size of the group and then to the probability that the 2 people selected will include Claire but not Max.
Sufficient.
(2) The probability that the 2 people selected will include neither Claire nor Max is \(\frac{2}{5}\).As is the case with statement (1), this statement will be true only of the group is of one particular size.
So, we could work from this probability to the size of the group and then to the probability that the 2 people selected will include Claire but not Max.
Sufficient.
Correct answer: D