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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.

I suppose 8 socks means, that there are 8 and not 8 pairs.

Note that we don't know if there is any # of black socks is the drawer. Let B be the # of black socks. And W the # of white socks.

(1) \(\frac{B}{8}<0.2\) --> \(B<1.6\), so there can be 1 or 0 black socks in the drawer. In any case as the # is less then 2 the probability of picking 2 black socks is 0. Sufficient.

(2) \(\frac{W}{8}>0.8\) --> \(W>6.4\), so there are 7 or 8 white socks in the drawer. As the maximum possible # of black socks is 1, thus the probability of 2 blacks is 0. Sufficient.

I had initially thought answer as D but later changed to A(my mistake). My thought process was that with second choice I don't know if remaining sock is black or not. I failed to calculate that the answer of two draws would be zero in this case also
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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.

Please explain in detail.

How about I tell you on what lines to think and perhaps you can arrive at the answer? Let's say there are x black socks in the drawer. When I pick the first one, the probability of picking a black sock is x/8 which is less than 1/5. Any ideas?
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25 Jun 2014, 05:06

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27 Jul 2014, 06:19

Hi Bunuel, Why is that in this problem the socks are assumed to be either white or black? No where in the question is it mentioned that the socks are either white or black.

Hi Bunuel, Why is that in this problem the socks are assumed to be either white or black? No where in the question is it mentioned that the socks are either white or black.

So, shouldn't the answer be E?

We are not assuming that.

From (1) we have that there could be 1 or 0 black socks in the drawer (for example, there could be 1 black and 7 red socks, or all red socks). No, matter which it is, the probability of picking 2 black socks is 0.

From (2) we have that there could be 7 or 8 white socks in the drawer (for example, there could be 7 white and 1 green, or 7 white and 1 black, or 8 white socks). So, there could be at most 1 black sock (0 or 1). So, in any case the probability of picking 2 black socks is 0.

The bottom line is that, from each statement we have that the maximum number of black socks is 1, which makes the probability of picking 2 black socks equal 0.

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20 Aug 2015, 01:51

Hello from the GMAT Club BumpBot!

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29 Aug 2016, 23:27

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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