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The Binomial Distribution

Main Concept

The binomial distribution is a discrete probability distribution that is used to obtain the probability of observing exactly k number of successes in a sequence of n trials, with the probability of success for all single trials of p. The binomial distribution describes a distribution where there are two mutually exclusive outcomes to an event. When n = 1, the binomial distribution is a Bernoulli distribution.

If a random variable X follows the binomial distribution, the probability of getting k successes in n trials is given by the following probability mass function:

fk = PX=k= nk pk 1p nk   


for k = 0,1,2, ... , n



nk = n!k!nk! 

p is probability of a successful event

k is the number of success trials

n is the number of trials


The cumulative distribution function is defined as:

fk = PXk=i= 0kni pi 1p ni   


for k = 0,1,2, ... , n


Note that the probability function and cumulative distribution function for the binomial distribution are only defined for integer values for k and as such, there is no continual curve which can be drawn through the point.  The gray lines are only for illustrating the shape of each function over the interval.


If n = number of trials, and p = probability of a successful event then:


nk pk 1p nk

The probability mass function


i= 0kni pi 1p ni

The cumulative distribution function



The expected value of a binomial random variable

Standard Deviation


Represented by the symbol σ, representing how much variation or spread exists from the mean value.




Suppose a biased coin comes up head with a probability of 0.2 when tossed. What is the probability of achieving 0, 1, 2, 3,and 4 heads after 4 tosses?


Let probability of success, p = 0.2, and number of trials, n = 4

P0 heads = f0 = PX= 0 = 40 0.20 10.2 40= 0.4096

P1 heads = f1 = PX= 1 = 41 0.21 10.2 41= 0.4096

P2 heads = f2 = PX= 2 = 42 0.22 10.2 42= 0.1536

P3 heads = f3 = PX= 3 = 43 0.23 10.2 43= 0.0256

P4 heads = f4 = PX= 4 = 44 0.24 10.2 44= 0.0016



See how the probability function of the binomial distribution changes with different values for the number of successes k, number of trials n, and the probability of success p.


# of successes (k) =

# of trials (n) =

Probability of success (p) =


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