The Binomial Probability Distribution

Chapter 4. Probability Distributions: Discrete Probability Distributions

The Binomial Probability Distribution

Binomial Experiment
In a binomial experiment:

We conduct #n# independent Bernoulli trials.
The probability of success #p# is the same for each trial.
The variable of interest #X# is the total number of successes observed.

Binomial Distribution
Let #X# be the number of successes among #n# trials in a binomial experiment, then #X# is a binomial random variable with range: \[R(X)=\{0,1,\ldots,n\}\]

We say that #X# is binomially distributed with parameters #n# and #p# and write this as: \[X \sim B(n,p)\]

GeoGebra

Suppose we flip a coin #12# times and define a success as the coin coming up Heads.

Let #X# be the number of successes among the #12# inquiries.

Then #X# is binomially distributed with #n=12# and #p=\mathbb{P}(\textit{Heads}) = 0.5#: \[X\sim B(12,0.5)\]

Suppose that #21\%# of people in a large population are smokers. Choose #100# people at random and ask them: "Are you a smoker"? Define a person answering "Yes" as a success.

Let #Y# be the number of successes among the #100# observations.

Then #Y# is binomially distributed with #n=100# and #p=0.21#: \[Y\sim B(100, 0.21)\]

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Computation of Binomial Probabilities with Statistical Software
Let #X# be a binomial random variable with parameters #n# and #p#.

To compute #\mathbb{P}(X=x)# in Excel, make use of the following function:

BINOM.DIST(x, n, p, cumulative)

x: The number of successes.

n: The number of trials.

p: The probability of success for each trial.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution (at most x successes), #\mathbb{P}(X \leq x)#

FALSE - uses the probability mass function (exactly x successes), #\mathbb{P}(X = x)#

To compute #\mathbb{P}(X=x)# in R, make use of the following function:

dbinom(x, size, prob)

x: The number of successes.

size: The number of trials.

prob: The probability of success for each trial.

Suppose #X# has a binomial distribution with #n=12# and #p=0.2#.

Compute #\mathbb{P}(X = 1)#. Round your answer to #3# decimal places.

#\mathbb{P}(X = 1)=0.206#

There are a number of different ways we can calculate #\mathbb{P}(X = 1)#. Click on one of the panels to toggle a specific solution.

Excel Calculation

To calculate #\mathbb{P}(X = 1)# in Excel, make use of the following function:

BINOM.DIST(x, n, p, cumulative)

x: The number of successes.

n: The number of trials.

p: The probability of success of each trial.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution (at most x successes), #\mathbb{P}(X \leq x)#

FALSE - uses the probability mass function (exactly x successes), #\mathbb{P}(X = x)#

Thus, to calculate #\mathbb{P}(X = 1)#, run the following command:
\[= \text{BINOM.DIST}(1, 12, 0.2, \text{FALSE})\]
This gives:
\[\mathbb{P}(X = 1) = 0.206\]

R Calculation

To calculate #\mathbb{P}(X = 1)# in R, make use of the following function:

dbinom(x, size, prob)

x: The number of successes.

size: The number of trials.

prob: The probability of success of each trial.

Thus, to calculate #\mathbb{P}(X = 1)#, run the following command:
\[\text{dbinom}(x = 1, size = 12, prob = 0.2)\]
This gives:
\[\mathbb{P}(X = 1) = 0.206\]

New example

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Computation of Cumulative Binomial Probabilities with Statistical Software
Let #X# be a binomial random variable with parameters #n# and #p#.

To calculate cumulative probabilities for a binomial distribution in Excel, make use of the following function:

BINOM.DIST(x, n, p, cumulative)

x: The number of successes.

n: The number of trials.

p: The probability of success for each trial.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution (at most x successes), #\mathbb{P}(X \leq x)#

FALSE - uses the probability mass function (exactly x successes), #\mathbb{P}(X = x)#

To calculate cumulative probabilities for a binomial distribution in R, make use of the following function:

pbinom(q, size, prob)

q: The number of successes.

size: The number of trials.

prob: The probability of success for each trial.

Cumulative Properties of Discrete Distributions
Since the binomial distribution is discrete, the following properties regarding its cumulative probabilities hold:

#\mathbb{P}(X \leq x) = \mathbb{P}(X=0) +\mathbb{P}(X=1) + \ldots + \mathbb{P}(X=x)#
#\mathbb{P}(X \lt x) = \mathbb{P}(X \leq x-1)#
#\mathbb{P}(X \gt x) = 1 - \mathbb{P}(X \leq x)#
#\mathbb{P}(X \geq x) = 1 - \mathbb{P}(X \leq x-1)#

Additionally, for any #2# values #a# and #b# in the range of #X# (with #a<b#):

#\mathbb{P}(a \leq X \leq b) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a - 1)#
#\mathbb{P}(a \leq X \lt b) = \mathbb{P}(X \leq b - 1) - \mathbb{P}(X \leq a - 1)#
#\mathbb{P}(a \lt X \leq b) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a)#
#\mathbb{P}(a \lt X \lt b) = \mathbb{P}(X \leq b - 1) - \mathbb{P}(X \leq a)#

Suppose #X# has a binomial distribution with #n=12# and #p=0.2#.

Compute #\mathbb{P}(X \leq 3)#. Round your answer to #3# decimal places.

#\mathbb{P}(X \leq 3)=0.795#

There are a number of different ways we can calculate #\mathbb{P}(X \leq 3)#. Click on one of the panels to toggle a specific solution.

Excel Calculation

To calculate #\mathbb{P}(X \leq 3)# in Excel, make use of the following function:

BINOM.DIST(x, n, p, cumulative)

x: The number of successes.

n: The number of trials.

p: The probability of success of each trial.

cumulative: A logical value that determines the form of the function.

TRUE - uses the cumulative distribution (at most x successes), #\mathbb{P}(X \leq x)#

FALSE - uses the probability mass function (exactly x successes), #\mathbb{P}(X = x)#

Thus, to calculate #\mathbb{P}(X \leq 3)#, run the following command:
\[= \text{BINOM.DIST}(3, 12, 0.2, \text{TRUE})\]
This gives:
\[\mathbb{P}(X \leq 3) = 0.795\]

R Calculation

To calculate #\mathbb{P}(X \leq 3)# in R, make use of the following function:

pbinom(q, size, prob)

q: The number of successes.

size: The number of trials.

prob: The probability of success of each trial.

Thus, to calculate #\mathbb{P}(X \leq 3)#, run the following command:
\[\text{pbinom}(q = 3, size = 12, prob = 0.2)\]
This gives:
\[\mathbb{P}(X \leq 3) = 0.795\]

New example

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Mean, Variance, and Standard Deviation of a Binomial Random Variable
Let #X# be a binomially distributed random variable with parameters #n# and #p#.

Then the expected value of #X# calculated with the following formula: \[\mu = n\cdot p\]

The variance of #X# is calculated with the following formula:\[\sigma^2 = n\cdot p \cdot (1-p)\]

And the standard deviation of #X# is calculated with the following formula:\[\sigma = \sqrt{n\cdot p \cdot (1-p)}\]

Let #X# be a binomially distributed random variable with parameters #n=20# and #p=0.25#.

Calculate the expected value of #X#. Round your answer to #2# decimal places.

#\mu=5.00#

The expected value of a binomial random variable #X\sim B(n,p)# is calculated as follows:
\[\begin{array}{rcl}
\mu&=&n \cdot p \\\\
&=& 20 \cdot 0.25 \\\\
&=& 5.00
\end{array}\]

New example