Probability question

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jchung402: Newbie | Next Rank: 10 Posts; Posts: 4; Joined: Tue Aug 30, 2011 3:57 pm

Probability question

by jchung402 » Wed Sep 17, 2014 7:40 am

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.

Here, am I only to assume there are only white and black socks?
B/c in another similar problem from the same source, the explanation said that we are not certain how many different colors of socks are in the drawer.
I am bit confused how I should approach this type of problem in the future.
Please advise.

Thank you

Quote

Source: Beat The GMAT — Data Sufficiency | Wed Sep 17, 2014 7:40 am

GMAT/MBA Expert

ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pm; Thanked: 1443 times; Followed by:247 members

by ceilidh.erickson » Wed Sep 17, 2014 8:00 am

You're right - you shouldn't assume that there are only black and white socks (what a boring sock drawer that would be!). You have to see if the statements limit it to just those possibilities.

If the statements told us that there was exactly a 0.2 chance of a black sock and 0.8 chance of a white sock on the first pick, then we would easily know that those are the only two options. If two probabilities add to 1, they have to be the only two options (assuming that they are mutually exclusive, as black and white socks are. A sock can't be both).

Here, we're told "less than 0.2" in the first statement and "more than 0.8" in the 2nd, which might not seem to be enough at first glance. However, we're also told that there are exactly 8 socks. So we have to consider the possibilities:

1 black, 7 non-black --> a 1/8 or 0.125 chance of picking black on the first pick
2 black, 6 non-black --> a 2/8 or 0.25 chance of picking black

Here, we see that we're already greater than a 0.2 probability, so it must be that there is only one black sock. So we know for sure that the probability of 2 black socks = 0. Sufficient.

If we consider the 2nd statement, ignore the fact that it says white socks, and just think of those as "non-black":

7 non-black, 1 black --> 0.875 chance of a non-black

Again, this is the only case that would give us a probability greater than 0.8 for non-black socks, so we must have exactly 1 black sock. Sufficient.

The answer is D.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education