DI Data Sufficiency | Numbers & Probability

A very interesting GMAT hard math data sufficiency question. A GMAT 675+ level DI question testing your understanding of multiple concepts – positive and negative numbers, computing probability, and interpreting the import of absolute values of two variables being the same.

Directions

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
EACH statement ALONE is sufficient to answer the question asked.
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Question

Set A contains distinct integers: A = {2, 4, 6, -8, x, y}. When two numbers from this set are picked and multiplied, what is the probability that the product is less than zero?
(1) x * y is not equal to zero.
(2) |x| = |y|

Correct Answer

Correct Answer is choice B. Statement 2 alone is sufficient.

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Explanatory Answer

Step 1: Decoding the Question Stem

Here is a summary and interpretation of what is given in the question stem

• Set A comprises 6 numbers, all of which are distinct integers. So, no two numbers will be same. x is not equal to y.
• Two numbers out of the 6 are picked at random and multiplied. The number of different product values that we will get is the number of different pairs of numbers one can pick from the set of 6 numbers. i.e., 6C2 = 15.
• The product of the numbers has to be less than zero. i.e., the product has to be negative. The product of two numbers is negative when one of the numbers is negative and the other number is positive.
• The question is asking us to compute the probability that the product of the selected two numbers is negative.
  • The numerator for the computation will be the number of outcomes in which the product will be negative.
  • The denominator for the computation will be the total number of products that we can get. In this question it is 15.
• What will help us find the answer? How many of the 6 numbers are positive and how many are negative?
  • Leaving x and y aside, 3 of the numbers are positive and 1 is negative

What does it all boil down to?

Can we determine whether x and y are positive or negative or zero using the information in the two statements.

Step 2: Evaluate Statement 1 Alone

Statement 1: x * y is not equal to zero.

Neither x nor y is zero. Though a valuable information, it still leaves more things unsaid. Is x or y positive and is x or y negative is not answered.

Statement 1 alone is therefore, not sufficient.

Eliminate answer options A and D.

Step 3: Evaluate Statement 2 Alone

Statement 2: |x| = |y|

This one is a very interesting statement.

If |x| = |y|, then we have the following inferences

1. x = y = 0
2. Both x and y are positive and have the same magnitude. For example x = y = 11
3. Both x and y are negative and have the same magnitude. For example x = y = -3
4. x and y are of opposite signs with the same magnitude. For example x = 3 and y = -3

Possibilities 1, 2 and 3, that both x and y are zero or both x and y are positive or both x and y are negative are not possible scenarios because the question stem states that the 6 numbers are distinct integers. So, x cannot be equal to y.

In possibility 4, we have one of x or y being positive and the other being negative. So, the set will comprise 4 positive numbers and 2 negative numbers.
The number of outcomes in which the product will be negative is when one of the two negative numbers is selected and one of the 4 positive numbers is selected. i.e., in 2 * 4 = 8 ways.

That being the case, the probability will be 8/15

Because we are able to find a unique answer to the question using statement 2, statement 2 alone is sufficient to answer the question.

Tip: We computed the exact probability in the above explanation. That step is not necessary as long as we can establish that we will get a unique value. You will easily save 45 seconds by skipping that step.

Choice B is the answer to the question.

Video Explanation