Tips & Documentation

The page replaces pages & pages of tables in your text. It provides more information in a more straightforward fashion.

Terminology

• n is the number of (Bernoulli) trials.
• p is the probability of success on each trial.
• X is the random variable that counts the number of successes in the n trials.
• x is used to represent a value X might take.
• P[ X = x ] is the probability that the number of successes is equal to x.
• P[ X <= x ] is the probability that the number of successes is at most (less than or equal to) X.

Input area

Each of the three boxes in the input area must have legal values.

• n must be a positive integer 1, 2, 3, . . .
• p must be above 0 and below 1.
• x must be between 0 and n.

Results area

Mean & standard deviation

The mean and standard deviation of a binomial variable, n trials and p probability, are given by

Exact probabilities

For every p between 0 and 1 and each n in 1, 2, . . ., 1000 the program will compute exact values (to the nearest 0.00001) for P[ X = x ] and P[ X <= x ]. Note that because X is a count, non-positive-integer values (-2, 0.34, 17.49, etc.) have 0 probability.

Normal approximations

Generally these are good approximations for situations that pass the "at least 10 test": both np and n(1 - p) are at least 10. However, this program will compute normal approximations for all legitimate input values. Notes:

You should not use these approximate values in place of the exact values. They may be close, but the exact values are the exact values.

You should not use these approximate values when the "at least 10 test" is failed.

When n is very large (> 1000 here) a huge amount of computing may be necessary to find exact probabilities; as a result the exact probabilities are not found for such cases. If the conditions pass the "at least 10 test" then the normal approximations should suffice.

If n > 1000 and the "at least 10 test" is failed, the program provides no useful information. One possible solution is the Poisson approximation (a feature I hope to include here someday). Ask a statistician!

Methods

Exact probabilities

For integer x between 0 and n inclusive

If x is non-integer the probability is automatically 0! So, for any x between 0 and 1:

(The "floor" operator rounds a number down.)

Normal Approximations

Denote standard normal probabilities by

For x between 0 and n inclusive

and for integer x between 0 and n inclusive

For non-integer values, this probability is automatically 0.

These approximations are continuity corrected.