The Binomial Distribution

4. Probability Distributions: Common Distributions

The Binomial Distribution

Binomial Setting

In a binomial setting:

There are $n$ independent observations
Each observation only has $2$ possible outcomes: 'success' or 'failure'.
The probability of a success, denoted $p$ , is the same for each observation.

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Binomial Distribution

Let $X$ denote the number of successes among $n$ observations in a binomial setting.

Then $X$ is a discrete random variable with $\{0, 1,2,\ldots,n\}$ as possible outcomes.

We say $X$ has the binomial distribution with parameters $n$ and $p$ , and we write $X \sim B(n,p)$ .

Flip a coin $12$ times and define a success as the coin coming up Heads.

Let $X$ be the number of successes among the $12$ observations.

Then $X$ is binomially distributed with $n=12$ and $p=\mathbb{P}(\textit{Heads}) = \frac{1}{2}$ , denoted $X\sim B(12,0.5)$

Suppose $21\%$ 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 $p=0.21$ , denoted $Y\sim B(100, 0.21)$

(This is approximately binomial because the total population is large relative to the sample size.)

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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 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), $\mathbb{P}(X = x)$

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 of each trial.

Suppose $X$ has a binomial distribution with $n=13$ and $p=0.8$ .

Compute $\mathbb{P}(X = 12)$ . Round your answer to $3$ decimal places.

$\mathbb{P}(X = 12)=0.179$

There are a number of different ways we can calculate $\mathbb{P}(X = 12)$ . Click on one of the panels to toggle a specific solution.

Excel Calculation

To calculate $\mathbb{P}(X = 12)$ 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 $\mathbb{P}(X = 12)$ , run the following command:
$= \text{BINOM.DIST}(12, 13, 0.8, \text{FALSE})$
This gives:
$\mathbb{P}(X = 12) = 0.179$

R Calculation

To calculate $\mathbb{P}(X = 12)$ 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 $\mathbb{P}(X = 12)$ , run the following command:
$\text{dbinom}(x = 12, size = 13, prob = 0.8)$
This gives:
$\mathbb{P}(X = 12) = 0.179$

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Cumulative Distribution

For any random variable $X$ , the cumulative distribution of $X$ is $\mathbb{P}(X \leq x)$ for any number $x$ .

Since binomial variables are discrete, the following properties regarding cumulative probabilities hold:

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

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

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

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 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), $\mathbb{P}(X = x)$

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 of each trial.

Suppose $X$ has a binomial distribution with $n=10$ and $p=0.5$ .

Compute $\mathbb{P}(X \leq 7)$ . Round your answer to $3$ decimal places.

$\mathbb{P}(X \leq 7)=0.945$

There are a number of different ways we can calculate $\mathbb{P}(X \leq 7)$ . Click on one of the panels to toggle a specific solution.

Excel Calculation

To calculate $\mathbb{P}(X \leq 7)$ 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 $\mathbb{P}(X \leq 7)$ , run the following command:
$= \text{BINOM.DIST}(7, 10, 0.5, \text{TRUE})$
This gives:
$\mathbb{P}(X \leq 7) = 0.945$

R Calculation

To calculate $\mathbb{P}(X \leq 7)$ 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 $\mathbb{P}(X \leq 7)$ , run the following command:
$\text{pbinom}(q = 7, size = 10, prob = 0.5)$
This gives:
$\mathbb{P}(X \leq 7) = 0.945$

New example