Chapter 7, Problem 7.23P

Urn I contains 5 white and 6 black balls, while urn 2 contains 8 white and 10 black balls. Two balls are randomly selected from urn 1 and are put into urn 2. If 3 balls are then randomly selected from urn 2, compute the expected number of white balls in the trio.

Hint: Let $X_i=1$ if the ith white ball initially in urn us one of the three selected, and let $X_i=0$ otherwise. Similarly, let $Y_i=1$ if the ith white ball from urn 2 is one of the three selected, and let $Y_i=0$ otherwise. The number of white balls in the trio can now be written as ${\displaystyle \underset{1}{\overset{5}{\sum }}{}_{X_i+}}{\displaystyle \underset{1}{\overset{8}{\sum }}{}_{Y_i}}$.