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.

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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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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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 $p=0.5$ .

Compute $\mathbb{P}(X \leq 4)$ . Round your answer to $3$ decimal places.

$\mathbb{P}(X \leq 4)=0.194$

There are a number of different ways we can calculate $\mathbb{P}(X \leq 4)$ . Click on one of the panels to toggle a specific solution.

Excel Calculation

To calculate $\mathbb{P}(X \leq 4)$ 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 $\mathbb{P}(X \leq 4)$ , run the following command:
$= \text{BINOM.DIST}(4, 12, 0.5, \text{TRUE})$
This gives:
$\mathbb{P}(X \leq 4) = 0.194$

R Calculation

To calculate $\mathbb{P}(X \leq 4)$ 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 $\mathbb{P}(X \leq 4)$ , run the following command:
$\text{pbinom}(q = 4, size = 12, prob = 0.5)$
This gives:
$\mathbb{P}(X \leq 4) = 0.194$

New example

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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