Breaking Down GMATPrep Probability Problems
Recently at ManhattanGMAT, weve been stuck on a certain theme: what kinds of characteristics make problems even harder? We looked at one example last week; this week, were going to tackle another: a seriously difficult problem from GMATPrep that is easiest to do if you really understand the theory behind the concept of probability. Sound like fun?
Heres the GMATPrep problem. Set your timer for 2 minutes. and GO!
* A certain jar contains only b black marbles, w white marbles, and r red marbles. If one marble is to be chosen at random from the jar, is the probability that the marble chosen will be red greater than the probability that the marble chosen will be white?(1) r/(b+w) >w/(b+r)
(2) (b-w) > r"
Theres a jar and it contains three kinds of marbles: black, white, and red. The yes/no question asks us whether the probability of choosing red is greater than the probability of choosing white. Im not a fan of probability and combinatorics in general, but when I first read this problem, I thought: okay, this doesnt sound so bad.
Then, I looked at the statements. Yeah. This is a tough one.
Our first task is to understand what the question is really asking, possibly in word format, possibly in algebraic format, or possibly both. From a word standpoint, if its true that the probability of choosing a red marble is greater than the probability of choosing a white marble, that must mean that there are more red marbles in the jar than white ones. If there are, then the answer to the question is yes. If there is an equal number of both, or if there are more white marbles, then the answer to the question is no. So sufficient information would allow me to tell either that (1) there are more red marbles than white (= definitive yes), or that (2) there is an equal number or more white marbles than red (= definitive no). My question becomes, simply, are there more red marbles than white in the jar?
Algebraically, this is how to write the concept described:
Isr/(b+w+r) >w/(b+r+w)?
The left-hand fraction (the probability of r) translates to: the number of red marbles divided by the total number of marbles. Similarly, the right-hand fraction (the probability of w) translates to: the number of white marbles divided by the total number of marbles.
For a moment, I got excited, because that statement looks a whole lot like statement 1. There are some key missing pieces, though, so Id have to figure out how to manipulate that to get what I want. That doesnt look like a lot of fun.
As a result, I decided that I was going to start with statement 2.
(2) b-w>r
This tells us that the number of black marbles minus the number of white marbles is more than the number of red marbles. What can we deduce from that?
The number of black marbles must be more than the number of red marbles, because b w must be equal to or smaller than b itself. What about the white and red marbles?
Try some numbers. Lets set b=10.
If w=2 and r = 7, then the inequality in statement 2 is true: 8 > 7. In this case, r is larger than w, so the probability of picking a red marble is greater than the probability of picking a white marble. Weve found one instance in which the answer is yes.
When testing numbers on yes/no data sufficiency questions, our next task is to see whether we can think of some numbers that will give us the opposite answer. We just found a yes instance, so now we want to try to find a no instance. What would I need to do in order to get a no answer, and am I allowed to do that?
A no answer would require w and r to be the same or w to be larger than r. Can we think of some numbers that would fit either one of those cases and that make statement 2 valid?
Lets keep b=10. Now, lets make w equal to 5 and r equal to 4. Is statement 2 valid? 10-5 > 4. 5 > 4. Yes, statement 2 is valid, so Im allowed to pick these numbers. Do I get a yes or no answer with these numbers? No! This time, w is larger than r, so the answer to the original question is no. This statement is insufficient; eliminate answers B and D.
Okay, now we have to move on to that ugly statement 1.
(1) r/(b+w) >w/(b+r)
We have several choices:
(1) Try to manipulate statement 1 algebraically to match the algebraic representation of our rephrased question.
(2) Try numbers again, as we did for statement 2.
(3) Think things through conceptually that is, use theory.
A conceptual approach is often the easiest and fastest as long as you fully understand the theory. If you dont, then a conceptual approach is definitely not the way to go.
In this particular problem, Im betting most of you already know exactly what you need to know in order to assess this statement conceptually, because the concept we actually need to understand is how fractions work! We dont need to use probability.
Statement 1 gives us two fractions on either side of an inequality sign. The numerators contain the two variables in question, r and w. The denominators in question both contain b added to the other variable, the one not in the numerator.
The inequality tells us that the first fraction is the larger one. In order for one fraction to be larger than another, what needs to be true in general?
Lets see. [pmath]1/2>1/3[/pmath]. In this case, the numerators are the same, but the denominator of [pmath]1/2[/pmath] (which is 2) is smaller than the denominator of [pmath]1/3[/pmath] (which is 3). As numbers get larger in the denominator, the fraction gets smaller.
Alternatively, [pmath]2/3>1/3[/pmath]. In this case, the denominators are the same, but the numerator of [pmath]2/3[/pmath] (which is 2) is larger than the numerator of [pmath]1/3[/pmath] (which is 1). As numbers get larger in the numerator, the fraction gets larger.
Finally, [pmath]2/3>1/4[/pmath]. In this case, the numerator of [pmath]2/3[/pmath] is the larger numerator, and the denominator of [pmath]2/3[/pmath] is the smaller denominator. The same rules hold: a larger numerator means a larger fraction, and a smaller denominator also means a larger fraction.
What does this mean? In order for one fraction to be larger than another, either the numerator has to be larger, or the denominator has to be smaller, or both. And we were told that this inequality is true:
r/(b+w) >w/(b+r)
The numerator of the larger fraction is r, and the numerator of the smaller fraction is w. So one possibility is that r > w.
The denominator of the larger fraction is (b + w), and the denominator of the smaller fraction is (b + r). So another possibility is that (b + w) < (b + r). (Remember to flip that sign! The denominator of the larger overall fraction would be the smaller of the two denominators!) Simplify that. The second possibility is really: w < r.
Either r > w, or w < r, or both. No matter what, weve just figured out that r > w. Statement 1 gives us a definitive yes answer the statement is sufficient.
The correct answer is A.
Key Takeaways for Fraction Theory Problems:
(1) If you can simplify a probability question such that you take the concept of probability out of the question entirely (as we did on this problem), you may be able to avoid dealing with probability for the rest of the problem. I would argue, in fact, that this isn't a probability question at all - it's really a theoretical fraction problem (notice that I changed the name to Fraction Theory in the Takeaways title above!).
(2) The GMAT often uses inequalities to disguise information on data sufficiency. Recognize different forms of inequalities that are designed to hide specific things; for instance, if you see <0 or >0, you should immediately think about positive versus negative. In this case, the form that we saw in statement 1 is one way to indicate how fractions work in general.
(3) If something looks like its going to take a lot of calculation or manipulation and youre not even sure how to get there, consider making a guess and moving on. On this problem, I doubt I would even have tried to manipulate statement 1 algebraically to try to match the probability I wrote based upon the question stem. If its that much work, chances are Im missing something, and theyre just getting me to waste my time and energy.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.