Chapter 16.4, Problem 15P

Suppose you have 2 boxes, 50 black pearls and 50 white pearls. You can mix the pearls up any way you like and put them all back into the 2 boxes. You don’t have to put the same number of pearls in each box, but each box must have at least 1 pearl in it. You will then be blindfolded, and you will get to open a random box and pick a random pearl out of it.
a. Suppose you put 25 black pearls in box 1 and 25 black pearls and 50 white pearls in box 2. Calculate the probability of picking a black pearl by imagining that you were going to pick a random box and a random pearl in the box a large number of times. In the ideal, what fraction of the time would you pick box 1? What fraction of those times that you picked box 1 would you pick a black pearl from box 1, in the ideal? On the other hand, in the ideal, what fraction of the time would you pick box 2? What fraction of those times that you picked box 2 would you pick a black pearl from box 2, in the ideal? Overall, what fraction of the time would you pick a black pearl, in the ideal?
b. Describe at least 2 other ways to distribute the pearlsthan what is described in part (a). Find the probability ofpicking a black pearl in each case.
c. Try to find a way to arrange the pearls so that theprobability of picking a black pearl is as large as possible.