Chapter 4 introduced some of the basic ideas of probability. This chapter builds on it in two main respects. First, if you think of the examples of Chapter 4, things started to get fairly messy, even with small numbers. Recall the example of rolling a die three times, and nding the probabilities of getting 0, 1, 2, or 3 1s. There were eight branches to our tree diagram; one branch each led to 0 and 3 successes; three branches each led to 1 and 2. Hopefully it occurred to you that the number of branches was doubling with each roll, and that such trees were going to get unwieldy really fast. Indeed that is the case. For example, suppose there were eight rolls, and we wanted to nd the probabilities of 0, 1, 2, 3, 4, 5, 6, 7, or 8 1s. There would be 256 branches to our tree. You certainly would not want to draw such a tree, let alone count how many branches lead to 5 1s. We need a more formal, systematic approach. You will learn it in this chapter.