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The Binomial Distribution
Section 15.8
 A binomial distribution is a discrete distribution defined by
two parameters:
 The number of trials, n
 The probability of success, p
The discrete random variable (X) is applied to sampling with
replacement. The probability distribution associated with this
variable is the binomial probability distribution. (For sampling
without replacement, the hypergeometric probability distribution is
the model used but is not a part of this course!)
X~B ( n, p) is written to indicate that the discrete random variable X
is binomially distributed. “ ~ “ reads “is distributed as”
Properties of a Binomial Experiment
 There is a fixed number of trials: n trials
 On each one of the n trials, there is only one of two possible
outcomes, labeled “success” or “failure” (Bernoulli Trials)
 Each trial is identical and independent
 On each of the trials, the probability of a success , p, is always
the same, and the probability of a failure, q = 1 – p, is also
always the same.
The binomial probability distribution
function
n r
nr
  p (1  p)
r
 For example, the probability function for X~B(6, o.4) means
that there are 6 trials and the probability of success is 0.4.
 6
x
6 x
P( X  x)    0.4 0.6
 x
 If X~B(5, 0.6), find P(X=4)
Example
 Suppose a spinner has 3 blue edges and 1 white edge. For
each spin we will get either a blue or a white edge. If we call
a blue result a “success” and a white result a “failure”, then we
have a binomial experiment.
 Let the random variable X be the number of “successes” or
blue results.
P(success)= P(failure)=
Consider twirling the spinner 3 times, n = 3.
Therefore, X = 0, 1, 2, or 3.