Statistical Inference: A Short Course

5.3 The Binomial Probability Distribution

While there are many discrete probability distributions, by far the most important one is the binomial distribution. To set the stage for a discussion of this distribution, we will need to cover two preliminary notions: (1) counting issues; and (2) the Bernoulli probability distribution.

Table 5.9 Weekly Lottery Prizes.

Number of Prizes	Dollar Amount
1	25,000
4	5,000
50	500
945	0

Table 5.10 Prize Probabilities.

X	f(X)
25,000	0.001
5,000	0.004
500	0.050
0	0.945
1.000

5.3.1 Counting Issues

We may generally view the concept of a permutation as an ordered set of objects. If any two of the objects are interchanged, we have a new permutation. For instance, the letters “a, b, c” form a particular permutation of the first three letters of the alphabet. If we interchange b and c, then “a, c, b” gives us another distinct permutation of these letters. More specifically, a permutation is any particular arrangement of r objects selected from a set of n distinct objects, r ≤ n.

What is the total number of permutations of r objects selected from a set of n distinct objects? To answer this, let us find

(5.7)

the number of permutations of n different objects taken r at a time, where

For instance, ...