GMAT Data Sufficiency - A Drawer Contains 8 Socks, and 2 Socks are Selected at Random Without Replacement

Question: A drawer contains 8 socks, and 2 socks are selected at random without replacement. What is the probability that both socks are black?

1. The probability is less than 0.2 that the first sock is black.
2.  The probability is more than 0.8 that the first sock is white.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.        
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.        
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.        
4. EACH statement ALONE is sufficient.        
5. Statements (1) and (2) TOGETHER are not sufficient.

‘A drawer contains 8 socks, and 2 socks are selected at random without replacement. What is the probability that both socks are black?’ is a GMAT Quantitative Reasoning topic. The GMAT quantitative reasoning section analyses the candidates' ability to solve mathematical, and quantitative problems and interpret graphic data. This section of the GMAT exam comprises 31 questions that need to be completed in 62 minutes. The GMAT Data Sufficiency question in this section comes with five options. This question type in GMAT Quantitative analyses candidates’ quantitative problems and identifies relevance with the data given.

Solution and Explanation 

There is only one approach to solving this problem.

Explanation:

The given case scenario states that there are 8 socks in a drawer and 2 socks are randomly selected without any replacement. Although there has been no mention of the presence of any black socks or the number of black socks, the probability of both selected socks being black needs to be identified.

To find the probability of selecting black socks, there are two cases that have been given to find whether they are sufficient for this scenario and prove their relevance.

Assuming that the 8 socks drawn from the drawer are 8 in number and not 8 pairs of socks, let the number of Black socks be B.

While one of the cases mentions the presence of white socks, let's say the probability of the socks being white is W.

In that case, the following can be evaluated-

For the first case stating about the probability is less than 0.2 for the first sock being black, it can be evaluated that
b/8 < 0.2 => B<1.6

Considering that B<1.6, so there can be 1 or 0 black socks in the drawer. In any case, as the B is less than 2 the probability of picking 2 black socks is 0. Hence, this case can be considered to be sufficient for proving the statement.

In terms of the second case stating that the probability is more than 0.8 for the first sock to be white, it can be identified that W/8 > 0.8 => W > 6.4

Considering that W > 6.4, it can be assumed that there are 7 or 8 white socks in the drawer. As it can be stated that the maximum number of black socks in the drawer could be 1 and the probability of 2 black socks is 0. Therefore, the second statement is also sufficient.

Correct Answer: D

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