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# 6.4: Binomial distribution and Normal Distribution

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## Discrete probability distributions

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 success,$$p$$ is always the same, and the probability of failure, $$q=1- p$$, is also always the same.
4. The expected value $$E(X)= np$$.

## 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.