This month’s “brain teaser” is actually a Data Sufficiency question, which is a question type that appears on the Data Insights section of the GMAT. These questions tend to be tricky and reward reasoning more than pure math ability!
The answer choices are below the question, but if you have never done a Data Sufficiency question, keep in mind that your goal is to determine whether the statements are sufficient (first alone, and then if not alone, together) to answer the question. You don’t necessarily need to know what the actual answer would be, just whether the statement or statements are sufficient to answer the question. This understanding is absolutely critical on the below question!
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 1/15.
(2) The probability that the 2 people selected will include neither Claire nor Max is 2/5.
A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) NEITHER ALONE NOR TOGETHER are the statements sufficient.
Answer:
For most people, it takes a little while to orient to what is going on in this question. But once it’s fully processed, it should be clear that we know that Claire and Max are part of a group and that the only thing we don’t know is how many additional people are in the group. If we knew that, we could answer the question. This is critical! We don’t need to actually calculate the probability or even know how! We just need to know how many additional people are in the group!
Let’s just process that for a second, because it is really the key to being able to answer this question quickly. If we know that Claire and Max are part of a group and we know how many additional people are in the group, how could we not be able to calculate the probability? What more information could we possibly acquire? We would already know everything that could be known relevant to this question, so it must be that knowing the additional number of people would be sufficient to calculate the probability. If this is obvious to you, great! But in doing this question with students, we have learned that it’s not always obvious to students!
Turning to the statements, we can apply a similar type of reasoning. If we know that the probability that Claire and Max will be selected is 1/15, that must tell us the number of additional people in the group. Think about it this way, if there was only 1 additional person in the group, the probability that both Claire and Max would be selected would be pretty high. If there were 2 additional people in the group, the probability would be a little bit lower. If there were 100 additional people in the group, the probability would be much, much lower. Therefore, it must be that 1/15 would yield the additional number of people.
Remember, it cannot be the case that no additional number of people would correspond to a probability of 1/15. The statements are facts! The probability really is 1/15! The only thing that we are concerned with is whether there would be more than one additional number of people that would yield a probability of 1/15. But for the reasons stated above, that cannot be the case. Each additional person added to the group is going to lower the probability of Claire and Max being selected. So, there can be only one number of additional people that will correspond to 1/15. Therefore, statement 1 is sufficient.
Clearly, the same reasoning can be applied to statement 2, just in reverse. If there were only 1 additional person, it would actually be impossible to select neither Clare nor Max. If there were 2 additional people, the probability that neither would be selected would be relatively low, since one of them (or both of them) would be fairly likely to be selected. The more people who are added to the group, the higher the probability becomes that neither will be selected. If there were 100 additional people in the group, the probability that neither would be selected would be much, much higher. Therefore, statement 2 is also sufficient. The answer is D.
This reasoning is not easy for most people to process, but the question can be answered very quickly and easily if you think through it in the above way. The alternative, to actually calculate and “solve” the statements, is much more difficult mathematically and takes much more time! As with most data sufficiency questions (and, indeed, most GMAT questions more generally), the key is reasoning over math!