How to find conditional probability

Take this problem as an example: 

5 white socks and 3 black socks are in a drawer. Stefan takes out two socks at random. Work out the probability that Stefan takes out two socks of the same colour without replacement. 

 

Solution

For Stefan to pick two socks of the same colour, he has to pick either two white socks (White AND White) OR two black socks (Black AND Black). 

In conditional probability, or indicates an addition of probabilities while and indicates multiplication of probabilities. Also, the term without replacement means that Stefan cannot put the sock he has picked back into the drawer before picking the next sock. The implication of this is that after every selection Stefan has one less sock to pick from the drawer.

 

Bearing this in mind, let us work out the answer to the question. First, let us list the terms in our equation.

P(B1) is the probability of picking a black sock first. P(B2) is the probablity of pcking a black sock second. P(W1) is the probability  of picking a white sock first and P(W2) is the probability of picking a white sock second. P(S) is the probability of picking the same colour of socks.

 

Remember, for Stefan to pick two socks of the same colour, he has to pick either two white socks (White AND White) OR two black socks (Black AND Black).  Re-writing this using our terms gives us the following:

 

P(S) = (P(W1) AND P(W2)) OR (P(B1) AND P(B2))

 

Changing AND to * and OR to + we get the following

 

P(S) = (P(W1)  P(W2)) + (P(B1)  P(B2))

 

P(S) = (5/8 * 4/7) + (3/8 * 2/7)

 

P(S) = (20/56) + (6/56)

 

P(S) = 26/56 

 

Reducing the fraction to the smallest terms, we get

 

P(S) = 13/28