+819 Suppose we have a bag with $10$ slips of paper in it. Eight slips have a $5$ on them and the other two have an $12$ on them.
(a) What is the expected value of the number shown when we draw a single slip of paper?
(b) What is the expected value of the number shown if we add one additional $12$ to the bag?
(c) What is the expected value of the number shown if we add two additional $12$'s (instead of just one) to the bag?
(d) How many $12$'s do we have to add to make the expected value equal to $10$?
(e) How many $12$'s do we have to add before the expected value is at least $11$?
