# Binomial Probabiliy Distribution

## Binomial Probability Distribution

In a Binomial Distribution we perform a fixed number, n, of independent Bernoulli Trials where the probability of obtaining a success each time is p and the probability of a failure each time is q = 1 - p, i.e. we are taking a sample of fixed size n from a population with replacement, so that the probability of a success each time is a constant p. Let the random variable X be the variable number of successes in the sample. In the app, adjust the parameters n and p to see the graph of the probability density (mass) function (pdf) in grey. The location of the mean is indicated by the red point and the standard deviation is indicated by the length of the green horizontal line segment. Note that there are n+1 bars above the whole numbers 0 to n. It is often true that in order to see the tallest bars, it is impossible to see some of the shorter bars. Use the checkbox to show an approximation of the graph of the cdf. In the spreadsheet on the right you can see the entire pdf and cdf tables. To illustrate probabilities manipulate the values of l and u. You can show middle, left tailed, and right tailed probabilities.

## Normal Approximation

If n is large then a Binomial Distribution can be modeled somewhat effectively with a Normal Distribution with the same mean and standard deviation. The pdf of this normal approximation can be graphed by checking its checkbox. Normal approximations for the left tail, middle, and right tail probabilities are given in the table. The first are P(x<l), P(l<x<u), and P(x>u) from the normal distribution. The second set are using a discrete correction computing P(x<l-0.5), P(l-.5<x>u+.5), and P(x>u+0.5).