2. Why do we need to learn the formula for the binomial probability distribution?

Using the formula repeatedly can be very tedious. To cut down on tedious calculations, most people will use a binomial table such as the one found in Appendix II of this book.

(a) However, there are many applications for which a table in the back of any book is not adequate. For instance, compute P1r 5 32 where n 5 5 and p 5 0.735 Can you find the result in the table? Do the calculation by using the formula.

List some other situations in which a table might not be adequate to solve a particular binomial distribution problem.

(b) The formula itself also has limitations. For instance, consider the difficulty of computing P1r  2852 where n 5 500 and p 5 0.6 What are some of the difficulties you run into? Consider the calculation of P(r 5 285). You will be raising 0.6 and 0.4 to very high powers; this will give you very, very small numbers. Then you need to compute C500,285, which is a very, very large number. When combining extremely large and extremely small numbers in the same calculation, most accuracy is lost unless you carry a huge number of significant digits. If this isn’t tedious enough, consider the steps you need to compute P1r  2852 5 P1r 5 2852 1 P1r 5 2862 1 p 1 P1r 5 5002 Does it seem clear that we need a better way to estimate P1r  2852? In Chapter 6, you will learn a much better way to estimate binomial probabilities when the number of trials is large.