The Shape of a Binomial Probabilty Distribution

The best way to understand the effect of n and p on the shape of a binomial probability distribution is to look at some histograms, so let's look at some possibilities.

n=10, p=0.2	n=10, p=0.5	n=10, p=0.8

Based on these, it would appear that the distribution is symmetric only if p=0.5, but this isn't actually true. Watch what happens as the number of trials, n, increases:

n=20, p=0.8	n=50, p=0.8

Interestingly, the distribution shape becomes roughly symmetric when n is large, even if p isn't close to 0.5. This brings us to a key point:

As the number of trials in a binomial experiment increases, the probability distribution becomes bell-shaped. As a rule of thumb, if np(1-p)≥10, the distribution will be approximately bell-shaped.