Page 153 - Statistics for Dummies

Chapter 8: Random Variables and the Binomial Distribution
However, depending on what you’re recording, situations originally having
more than two outcomes can fall under the binomial category. For example, if
you roll a fair die 10 times and each time you record whether or not you get a
1, then condition 2 is met because your two outcomes of interest are getting
a 1 (“success”) and not getting a 1 (“failure”). In this case, p (the probabil-
5
1
ity of success) = ⁄6, and ⁄6 is the probability of failure. So if X is counting the
number of 1s you get in 10 rolls, X is a binomial random variable.
Trials are not independent
The independence condition is violated when the outcome of one trial
affects another trial. Suppose you want to know opinions of adults in your
city regarding a proposed casino. Instead of taking a random sample of, say,
100 people, to save time you select 50 married couples and ask each of them
what their opinion is. In this case it’s reasonable to say couples have a higher
chance of agreeing on their opinions than individuals selected at random, so
the independence condition 4 is not met. 137
Probability of success (p) changes
You have 10 people — 6 women and 4 men — and you want to form a commit-
tee of 2 people at random. Let X be the number of women on the committee
of 2. The chance of selecting a woman at random on the first try is ⁄10. Because
6
you can’t select this same woman again, the chance of selecting another
5
woman is now ⁄9. The value of p has changed, and condition 3 is not met.
If the population is very large (for example all U.S. adults), p still changes
every time you choose someone, but the change is negligible, so you don’t
worry about it. You still say the trials are independent with the same probabil-
ity of success, p. (Life is so much easier that way!)
Finding Binomial Probabilities
Using a Formula
After you identify that X has a binomial distribution (the four conditions from the
section “Checking binomial conditions step by step” are met), you’ll likely want
to find probabilities for X. The good news is that you don’t have to find them
from scratch; you get to use established formulas for finding binomial probabili-
ties, using the values of n and p unique to each problem. Probabilities for a
binomial random variable X can be found using the following formula for p(x):

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