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( ) ( )
We rewrite our formula with the above defined components:
. / ( )
This is known as the binomial probability density function, or binomial pdf.
To make this more clear, we first define a random variable, . In the case of a binomial
experiment (one in which there are two possible outcomes for each trial), the set listing all
possible values that can be achieved (between 0 and the number of trials).
For example, if
in coin tosses, then * +. That is, between 0 and 10 heads can possibly
be achieved in 10 tosses of the coin (though not all have the same probability). To indicate a
binomial pdf calculation, we often write:
The probability that takes on successes is . / ( ) , or,
( ) . / ( )
We summarize a binomial pdf below, along with the necessary assumptions to use this.
Binomial Probability Density Function (pdf)
If the following assumptions are met:
1) An experiment is carried out with trials,
2) Each trial can result in only one of two possible values: a success or a failure,
3) The probability of a success in each trial is (it is always the same), and
4) Each trial is independent of all other trials (the outcome of one trial in no way affects the
outcome of any other trial),
then the experiment is a binomial experiment and the probability of
calculated by
successes can be
( ) . / ( )
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