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

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Fri May 01, 2020 4:57 pm

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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.

OA D

Source: Magoosh

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Source: Beat The GMAT — Data Sufficiency | Fri May 01, 2020 4:57 pm

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

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

by deloitte247 » Sun May 03, 2020 5:31 am

Let the number of black socks = b
Let the number of white socks = w

Target question => What is the probability that both socks are black?

Statement 1 => The probability is less than 0.2 that the first sock is black
$$\frac{b}{8}<0.2\ and\ b<0.2\cdot8$$
$$b<1.6$$
Since the total number of black socks is not up to 2, there is no way 2 selected socks can be black. It is either 1 or 0, therefore, you can't select from what does not exist. Statement 1 is SUFFICIENT since the probability of picking 2 black socks = 0

Statement 2 => The probability is more than 0.8 that the first sock is white
$$\frac{w}{8}>0.8\ and\ w>6.4$$
The total number of white socks is greater than 6 i.e it is either 7 or 8. So, the number of black socks is either 0 or 1 and the probability of picking 2 black socks is 0 because the number of black socks is not even up to 2
Statement 2 is SUFFICIENT

Answer = D

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