A marble is drawn at random Whatever marble is drawn is retu

A marble is drawn at random. Whatever marble is drawn, is returned to the urn, along with two additional marbles of the same color. Another marble is then drawn from the resulting urn. What is the probability that the second marble drawn is blue?

Solution

The probability that the second marble is blue is the sum of the three disjoint events probabilities

P(red then blue) + P(white then blue) + P(blue then blue).

We have

P(red then blue)=(5/12)(3/14)
P(white then blue)=(4/12)(3/14)
P(blue then blue)=(3/12)(5/14)

Add them up and get

P=(5*3+4*3+3*5)/(12*14)=42/168=1/4