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Chapter 13. Binomial Distributions
The Binomial Setting and Binomial Distributions
Note. The binomial setting consists of an experiment with observations satisfying:
- There are a fixed number n of observations.
- The n observations are all independent. That is, know- ing the result of one observation does not change the prob- abilities we assign to other observations.
- Each observation falls into one of just two categories, which for convenience we call “success” and “failure.”
- The probability of a success, call it p, is the same for each observation.
Definition. The count X of successes in the binomial setting has the binomial distribution with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n.
Example S.13.1. Binomial Stooges. A TV station airs 10 (not necessarily different) stooge films per week. The films are chosen at random from the collection of 190 films. Since there are 97 films with Curly as the third stooge, the probability of a Curly film being chosen at random is p = 97/190. Therefore the count of the number of Curly films shown by the TV station follows a binomial distribution with parameters p = 97/190 and n = 10.
Binomial Distributions in Statistical Sampling
Note. Choose a simple random sample of size n from a pop- ulation with proportion p of successes. When the population is much larger than the sample, the count X of successes in the sample has approximately the binomial distribution with parameters n and p. This property will allow us to answer binomial questions with normal approximations.
Example S.13.2. Binomial Stooges Again. In example S.13.1, we considered a binomial experiment with p = 97/190 and n = 10. Make a table of P (X = k) for k = 0, 1 , 2 ,... , 10. Solution. Using the formula
P (X = k) =
n k
pk(1 − p)n−k,
we get: k 0 1 2 3 4 5 6 7 8 9 10 P (X = k) 0.0008 0.0082 0.0386 0.1075 0.1962 0.2455 0.2134 0.1272 0.0498 0.0115 0.
Binomial Mean and Standard Deviation
Note. If a count X has the binomial distribution with num- ber of observations n and probability of success p, the mean and standard deviation of X are μ = np and σ =
np(1 − p).
Example. Exercise 13.9 page 334.
The Normal Approximation to Binomial Distributions
Note. Suppose that a count X has the binomial distribu- tion with n observations and success probability p. When n is large, the distribution of X is approximately the normal distribution N (np,
np(1 − p)). As a rule of thumb, we will use the normal approximation when n is so large that np ≥ 10 and n(1 − p) ≥ 10.
Figure 13.3 page 335. A histogram of 1000 trials of a binomial experiment with n = 2500 and p = 0.6.
