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Billy's drawer contains 5 blue socks and 6 black socks.

Billy pulls out two socks at random without replacement.

If the first sock drawn is black, what is the probability the second sock is also black and Billy has a pair?
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The question is a bit vague but will try and asnwer it to the best of my ability.

There are 11 socks i.e. 5 blue socks and 6 black socks. 2 socks were taken out without replacement.

So the probabilities are as follows:

\(P(\text{1st sock is blue} = \frac{5}{11})\)

\(P(\text{1st sock is black} = \frac{6}{11})\)

\(P(\text{2nd sock is blue given 1st is blue} = \frac{4}{10})\)

\(P(\text{2nd sock is black given 1st is blue} = \frac{6}{10})\)

\(P(\text{2nd sock is blue given 1st is black} = \frac{5}{10})\)

\(P(\text{2nd sock is black given 1st is black} = \frac{5}{10})\)

 

To answer the questions as I understand them.

 

a) What is the probability the second sock is also black given the first one is black?

\[P(\text{2nd sock is black given 1st is black} = \frac{5}{10}\]

 

b) What is the probability that both socks are black?

\[P(\text{Black and Black})=\frac{6}{11}\cdot \frac{5}{10}=\frac{30}{110}\]

 

 

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