Picking the Correct Distribution
Binomial, Negative Binomial, Geometric, or Hypergeometric?
Students often become confused when trying to decide whether a random variable in a
word problem fits a binomial distribution, negative binomial, geometric or
hypergeometric. This paper will explain the similarities and differences between these
four related distributions.
First, a binomial random variable must have
n
independent trials, they must be Bernoulli
trials (i.e., two choices only.
..1 or 0, heads or tails, yes or no, etc.), and the probability of
a "success" must be the same on each trial. We call the probability of success
p
. In order
for the probability of success
p
to be constant, this means that we are sampling from a
very large population, so that picking one sample does not materially affect the
probability of success on the next sample. An example of a very large population would
be the population of all fish in Lake Superior. The population is large enough that taking
a few fish out of the lake, even without replacing them, does not materially affect the
probabilities on the next pick.
A binomial random variable can tell us the probability of obtaining
k
successes out of
n
trials. For example, if we pick 20 people out of a large population, and we know that
there is a probability of 40% that any given member of the population smokes, we can
define a random variable
X
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 Spring '11
 Berg
 Binomial, Probability, Probability theory, Binomial distribution, Discrete probability distribution, binomial random variable, Hypergeometric

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