Graphs of binomial distribution using a graphics calculator (Investigation)

The Graph of the Binomial Distribution

We'll begin by interacting with the applet below to get a feel for how different values of $\displaystyle n$ and $\displaystyle p$ affect the distribution of our probabilities for a Binomial distribution.

Begin by setting the applet to $\displaystyle n=10$ and $\displaystyle p=0.5$.

How would you describe the distribution of the graph you see?

Remember when describing the shape of a histogram we use the phrases positively skewed, symmetrical and negatively skewed.

Keeping $\displaystyle p=0.5$, change the value of $\displaystyle n$. Does your description of the distribution stay the same?

In both cases, with $\displaystyle p=0.5$, you should see that the graphs are symmetrical.

This makes a lot of sense! A value of $\displaystyle p=0.5$ indicates an equal probability of success and failure, so you'd expect symmetry.

Now set the applet to $\displaystyle n=10$ and and slide the $\displaystyle p$ value to the left and to the right.

Questions:

1. As the probability of success decreases, what happens to the shape of the distribution?
2. As the probability of success increases, what happens to the shape of the distribution?

Answers:

1. The distribution becomes positively skewed with a tail to the right
2. The distribution becomes negatively skewed with a tail to the left

Now slide the $\displaystyle n$ and $\displaystyle p$ values around and confirm that those findings about the shape of the distribution hold for all scenarios.

Graphing the Binomial Distribution Using Your Graphics Calculator

Let's look through a series of screenshots to do these problems using the TI-Nspire.

Firstly through the menu we select Statistics and Distribution and select BinomialPdf to calculate $\displaystyle P(X=5)$ when $\displaystyle n=8$ and $\displaystyle p=0.3$

To calculate the cumulative probability of $\displaystyle P(1<=X<=3)$ we select BinomialCdf instead.