Data Sufficiency

danjuma wrote: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 < 0.2 that the first sock is black.

2. The probability is more than 0.8 that the first sock is white.

Can one use the probability formula for this type of question?

Thank you

Got a PM to reply on this thread.

Total no. of socks in the drawer = 8
Since we do not how many black socks are there, so let the no. of black socks = B
Similarly, let the no. of white socks = W

(1) Probability of picking a black sock = B/8
Now it is given that B/8 < 0.2 or B < 1.6, which means that either there is 1 black socks or 0 black sock. In either case, probability of picking 2 black socks = 0; SUFFICIENT to answer the question.

(2) Probability of picking a white sock = W/8
So, W/8 > 0.8 or W > 6.4, which implies there can be either 7 or 8 white socks in the drawer. If there are 7 white socks, then there is only 1 black socks and in case there are 8 white socks, then the black sock is 0. So, in either case the required probability is again 0; SUFFICIENT to answer the question.

The correct answer is D.

Hi jaspreetsra,

In this types of probability-based DS questions, it's important to be a little cynical. Ask yourself "what was I really told?", be prepared to take lots of little notes (stay organized) and be prepared to go to extremes.

Here, we're told that there are 8 socks and that 2 are selected at random without replacement. We're asked for the probability that they're BOTH BLACK. To start, we do NOT know what color the 8 socks are. Maybe they're all 1 color, maybe they're 2 colors,....maybe they're 8 different colors...

Now, imagine if there are...
NO black socks at all, then the probability is 0%
JUST 1 black sock, then the probability is still 0%
ALL black socks, then the probability is 100%

2 or more black socks will require us to do an actual calculation.

Fact 1: The probability is LESS than .2 (meaning less than 20%) that the first sock is black

Here, we have to focus on pulling that 1st sock.

0/8 = 0
1/8 = .125
2/8 = .25
3/8 = .375
Etc.

Since the probability is LESS than .2, there are only 2 possibilities:
1 black sock....so the probability of pulling 2 black socks is 0%
or
0 black socks...so the probability of pulling 2 black socks is 0%

Under these conditions, the probability is ALWAYS 0%
Fact 1 is SUFFICIENT

Fact 2: The probability is MORE than .8 that the first sock is white.

This is essentially the same information that we had in Fact 1:

6/8 = .75
7/8 = .875
8/8 = 1

This tells us that 7 OR 8 of the socks are white (meaning they are NOT black)
7 white socks --> 0 or 1 BLACK socks --> probability of pulling 2 black socks = 0%
8 white socks --> 0 BLACK socks --> probability of pulling 2 black socks = 0%

Under these conditions, the probability is ALWAYS 0%
Fact 2 is SUFFICIENT

Final Answer: D

GMAT assassins aren't born, they're made,
Rich