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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?
We need to find the probability that exactly one specific person (Claire) is selected while another specific person (Max) is not selected when choosing 2 people randomly from a group.
The question asks for a specific probability value. This is a value question - we need to determine whether we can find exactly one numerical answer for P(Claire selected AND Max not selected).
Here's the crucial insight: In any probability problem involving selections from a finite group, if we know the total group size, we can calculate any specific selection probability. The question becomes: Can we determine the group size from the given information?
Think of it this way - if we know there are n people total:
- We can count how many ways to select 2 people (all possible pairs)
- We can count how many ways to select Claire with someone other than Max
- The probability is simply the ratio of these two counts
So this question really boils down to: "Can we figure out how many people are in the group?"
Statement 1: The probability that the 2 people selected will be Claire and Max is \(\mathrm{1/15}\).
We know that selecting the specific pair {Claire, Max} happens with probability \(\mathrm{1/15}\). Since there's only ONE way to select this exact pair, this means the total number of possible pairs must be 15.
Let's think through what this means:
- If there are n people total, there are \(\mathrm{n(n-1)/2}\) possible pairs
- We're told this equals 15
- So \(\mathrm{n(n-1)/2 = 15}\)
This is a specific constraint that will have only one valid solution for n (since n must be a positive integer representing the number of people). Once we know n, we can determine exactly how many ways Claire can be paired with someone other than Max, and thus calculate our desired probability.
Statement 1 gives us enough information to uniquely determine the group size, which means we can calculate the exact probability we're looking for.
[STOP - Statement 1 is SUFFICIENT!]
This eliminates choices B, C, and E.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The probability that the 2 people selected will include neither Claire nor Max is \(\mathrm{2/5}\).
This tells us that \(\mathrm{2/5}\) of all possible pairs exclude both Claire and Max. In other words, when we select 2 people from the entire group, there's a \(\mathrm{2/5}\) chance that neither of them is Claire or Max.
Consider what this constraint tells us:
- If the group has n people total, then (n-2) people are neither Claire nor Max
- The number of ways to select 2 people from these (n-2) others is \(\mathrm{(n-2)(n-3)/2}\)
- The total number of ways to select 2 from n people is \(\mathrm{n(n-1)/2}\)
- We're told this ratio equals \(\mathrm{2/5}\)
This gives us the equation: \(\mathrm{[(n-2)(n-3)/2] ÷ [n(n-1)/2] = 2/5}\)
Just like with Statement 1, this is a specific constraint on n. This equation will determine a unique value for the group size (since n must be a positive integer). And once we know the group size, we can calculate any selection probability, including \(\mathrm{P(Claire\ selected,\ Max\ not\ selected)}\).
Statement 2 also uniquely determines the group size, making it sufficient to answer our question.
[STOP - Statement 2 is SUFFICIENT!]
Since each statement alone gives us enough information to determine the group size - and therefore to calculate the desired probability - each statement alone is sufficient.
Answer Choice D: "Each statement alone is sufficient."