1 Answer

I just read up on binomial distribution, and all I can say is, HUH?

In many cases, it is appropriate to summarize a group of independent observations by the number of observations in the group that represent one of two outcomes. For example, the proportion of individuals in a random sample who support one of two political candidates fits this description. In this case, the statistic is the count X of voters who support the candidate divided by the total number of individuals in the group n. This provides an estimate of the parameter p, the proportion of individuals who support the candidate in the entire population.
The binomial distribution describes the behavior of a count variable X if the following conditions apply:

1: The number of observations n is fixed.
2: Each observation is independent.
3: Each observation represents one of two outcomes ("success" or "failure").
4: The probability of "success" p is the same for each outcome.

If these conditions are met, then X has a binomial distribution with parameters n and p, abbreviated B(n,p).
Example

Suppose individuals with a certain gene have a 0.70 probability of eventually contracting a certain disease. If 100 individuals with the gene participate in a lifetime study, then the distribution of the random variable describing the number of individuals who will contract the disease is distributed B(100,0.7).

Note: The sampling distribution of a count variable is only well-described by the binomial distribution is cases where the population size is significantly larger than the sample size. As a general rule, the binomial distribution should not be applied to observations from a simple random sample (SRS) unless the population size is at least 10 times larger than the sample size.

To find probabilities from a binomial distribution, one may either calculate them directly, use a binomial table, or use a computer. The number of sixes rolled by a single die in 20 rolls has a B(20,1/6) distribution. The probability of rolling more than 2 sixes in 20 rolls, P(X>2), is equal to 1 - P(X<2) = 1 - (P(X=0) + P(X=1) + P(X=2)). Using the MINITAB command "cdf" with subcommand "binomial n=20 p=0.166667" gives the cumulative distribution function as follows:

Binomial with n = 20 and p = 0.166667

x P( X <= x)
0 0.0261
1 0.1304
2 0.3287
3 0.5665
4 0.7687
5 0.8982
6 0.9629
7 0.9887
8 0.9972
9 0.9994