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What A Binomial Random Variable’s Probability
Distribution Looks Like
1. Review.
You get a Binomial random variable when you have n independent trials, with p
chance of success each trial. The Binomial random variable is the x, the number
of successes out of n trials.
Here are some probability distributions of some Binomial random variables:
n=4
p = 0.5
n = 10
p = 0.5
They are mostly Normal shaped!! That is, if n isn’t small, and p isn’t too extreme.
Notice that if n is too small, or p is too small or too big, it doesn’t look Normal.
You can approximate the Binomial distribution with a Normal distribution as long
as np ≥ 10 and n(1 – p) ≥ 10.
Cool! What do we need to do Normal distribution problems?
Well, we need to be able to find z-scores. So we need µ and σ.
But when we have a Binomial situation, we have µ and σ. What are the formulas?
  np and   np(1  p) .
(Where did those come from?)
You can approximate the Binomial distribution with a Normal distribution as long
as np ≥ 10 and n(1 – p) ≥ 10. Use   np and   np(1  p) .
Example: 80 percent of people who flip off Mr. Simon lose their middle finger.
If you pick 100 people at random, use the Normal approximation to the Binomial
to find the probability that more than 75 will lose their finger.
1)
2)
3)
4)
Check that the approximation is legal.
Find the mean and standard deviation.
Find the z-score you need.
Use the calculator to find the approximate probability.
Approximate answer: .8944
Exact answer: 1 – binomcdf (100, .8, 75) = .8686
Not bad. Notice that we came pretty close to missing one of our checks, though.
But Mr. Simon you might ask, why would I do this when I can find the answer
exactly?
Theory, my good students! Theory!! It will be quite useful later on.
Try another one: Use the Normal approximation to the Binomial to estimate the
probability that less than 85 will lose their finger.