Now that we know a little about Bernoulli Trials and the Binomial Distribution, let's take a more visual look at the distribution and making the most of using it on our CAS calculators.
The Graph of the Binomial Distribution
We'll begin by interacting with the applet below to get a feel for how different values of n and p affect the distribution of our probabilities for a Binomial distribution.
n is the number of trials of a Bernoulli experiment (an experiment with only two outcomes, a success or a failure)
p is the probability of success of each trial and each trial is independent.
Begin by setting the applet to n=10 and 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 p=0.5, change the value of n. Does your description of the distribution stay the same?
In both cases, with p=0.5, you should see that the graphs are symmetrical.
This makes a lot of sense! A value of p=0.5 indicates an equal probability of success and failure, so you'd expect symmetry.
Now set the applet to n=10 and and slide the p value to the left and to the right.
Questions:
- As the probability of success decreases, what happens to the shape of the distribution?
- As the probability of success increases, what happens to the shape of the distribution?
Answers:
- The distribution becomes positively skewed with a tail to the right
- The distribution becomes negatively skewed with a tail to the left
Now slide the n and p values around and confirm that those findings about the shape of the distribution hold for all scenarios.
Graphing the Binomial Distribution Using Your CAS 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 P(X=5) when n=8 and p=0.3
To calculate the cumulative probability of P(1<=X<=3) we select BinomialCdf instead.
