Using The 5 lb. Book To Study Advanced Quant
I’ve got another one for you from our 5 lb. Book of GRE Practice Problems, and this one’s serious. I took it from the Advanced Quant chapter. Try it out and then we’ll chat!
Triplets Adam, Bruce, and Charlie enter a triathlon. There are nine competitors in the triathlon. If every competitor has an equal chance of winning, and three medals will be awarded, what is the probability that at least two of the triplets will win a medal?
(A) 3/14
(B) 19/84
(C) 11/42
(D) 15/28
(E) 3/4
© ManhattanPrep, 2013
Yuck. I’m not a fan of probability in general and this one is particularly annoying. Why? Because they ask for the probability that at least two will win. Most of the time, when a probability question uses at least or at most language, we can use the cool 1 “ x shortcut because there’s only one not-included case.
For example, if I tell you I’m going to flip a coin three times, I might ask you to calculate the probability that I’ll get at least one heads. There’s only one case where I wouldn’t: zero heads. So you can just calculate the probability of zero heads and subtract from 1.
But we can’t do that here, because it’s possible for just 1 twin to win a medal and it’s also possible for zero twins to win a medal. Sigh.
Okay, how are we going to tackle this? Probability is a measure of the number of desired outcomes divided by the total number of possibilities. Let’s figure out the total number of possibilities first.
Take a look at the question again. Is this one of those questions where the order matters? If you don’t win, you don’t win. If you do win, does the question make a distinction between coming in first, second, or third?
Nope. An actual contest probably would, but this question only makes one distinction: you either win a medal or you don’t. If Adam, Bruce, and Charlie come in 1-2-3, or if Charlie, Bruce, and Adam come in 1-2-3, the outcome is the same as far as the problem is concerned: all three win a medal. As a result, order doesn’t matter in this problem.
There are nine competitors; 3 will win a medal and 6 will not. Use the order doesn’t matter formula to figure out how many possible combinations there are:
And simplify:
3 × 4 × 7 = 84
Okay, there are 84 possible ways for 3 out of the 9 competitors to win medals.
Now, how many of those will include at least 2 of the triplets? Let’s start with the easier case: all 3 win. There’s actually only 1 possible way that all three triplets could take all 3 medals(since order doesn’t matter). All that matters is that the three medals are taken by the three brothers, in any order.
What about the cases where only two of the three win? Let’s stay that Adam and Bruce win but Charlie doesn’t. Is there only one way for that to happen?
No “ there are actually 6 ways for that to happen! Let’s say that the 6 other contestants are named D, E, F, G, H, and I, respectively. So one winning set could be Adam, Bruce, and D. Another winning set could be Adam, Bruce, and E. Essentially, Adam and Bruce could be paired with any one of the other 6 “ just not their brother Charlie.
Likewise, if Adam and Charlie are the two who win, they can be paired with any one of the other 6. And if Bruce and Charlie are the winners, they can also be paired with any one of the other 6.
There are 18 different ways, then, in which two of the three triplets win. Add that to the 1 way that all three can win: there are 19 different ways that at least two of the three triplets win.
Because there are 84 possible combinations overall, the probability that at least two of the triplets will win is 19/84.
The correct answer is B.
What if you had to guess? Is it more likely that all 3 of the triplets will win a medal or that none of the 3 will? Since there are more competitors, the competitors are more likely to take the top 3 spots than are the triplets.
Is it more likely that 2 of the 3 triplets will win a medal or that 1 of the 3 will? Let’s say that 1 triplet wins a medal. For the remaining two medals, we have a choice of 2 other triplets and 6 non-triplet competitors. It’s more likely, then, that the non-triplet competitors will win the remaining medals¾there are more of them. So, again, it’s more likely that a lower number of triplets will get medals.
The overall probability for at least 2 triplets, then, should be less than 50%. Eliminate answers D and E and guess from among the other three choices.
Key Takeaways for Messy Probability:
(1) First, probability isn’t super-common on the test, so if you hate these, do know the basics but also know how to guess. If you get a harder one (like this one), just guess and move on!
(2) If you do want to tackle a tough probability problem, first try to think about it in real-world terms. Pretend you’re the one running the triathlon and awarding the medals. This will make it easier to figure out, for example, that order doesn’t matter in this particular problem.
(3) Next, break things into smaller parts. Probability = desired / total. I have a formula to calculate the total. I can break the desired portion down into exactly-2-win or exactly-3-win. And so on!
© ManhattanPrep, 2013