
# 6.4: Binomial distribution and Normal Distribution

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

#### Discrete probability distribution

Binomial Trials

1. There are a fixed number of independent trials $$n.$$

2. Each trial has only two (hence binomial) outcomes, either “success” or “failure”.

3. For the trials, the probability of a success,$$p$$, is always the same, and the probability of failure, $$q=1- p$$, is also always the same.

#### Excel Activity

Goal:  Get a “feel” for binomial distributions by finding their probability distribution tables and graphing them.

Calculate the probability distribution table for X, a binomial distribution with 10 trials and probability of success p = 0.02.  Use the drag feature to save yourself from a lot of typing!

 X P(X = x) 0 (say this is in cell A2) =BINOMDIST (A2, 10,0.2,False) 1 2

Use Chart Wizard to plot the probabilities as a histogram (bar chart with no gaps!)  You’ll need to click on the bars of the chart and Select Data to get the 0, 1, 2, … as the X-axis labels and you’ll need to Select Format Data Series to remove gaps.

Repeat for n=10, p= 0.5  and n=10, p = 0.9.  You’ll get 3 tables and 3 histograms.  What are the shapes of each distribution?

Answer the following:   For small n Binomial Histograms tend to be ______ skewed if p < 0.5 and ______   skewed if p > 0.5.