Binomial distributions

This is about probability of having exactly, at most, or at least $k$ successes in $n$ trials.

When to Use
Definition
Calculator
Expected value
Variance and standard deviation
Mode and median
Tips

When to Use

If it exhibits the following symptoms, binomial distribution may be for you

for number of times something happened out of $n$ tries
each value or measurement is independent from the rest

For picking items from a large pool (in the ~thousands), selection without replacement can still be approximate by binomial distribution.

Definition

Random variable $X$ is binomially distributed is notated as

X \sim \text{B}(n, p)

with $n$ trials or attempts, and probability $p$ of success for each trial. It has the probability distribution

\text{P}(X = x) = \binom nx p^x(1-p)^{n-x}

where $\displaystyle \binom nx$ is the binomial coefficient or $^nC_x$ .

This formula is not on the formula booklet. But it is very similar to the one for binomial expansion.

Note: In $n$ tries, $x$ could take on $n + 1$ values from $0$ to $n$ .

Binomial distribution is about defining the success for a single trial, and treating different successes separately.

Calculator

All approved graphing calculators support probability $\text{P}(X = x)$ and cumulative probability $\text{P}(X \leq x)$ .

Some calculators also allow an interval of probability, such as $\text{P}(3 \leq X \leq 6)$ . Others do not, and would require doing $\text{P}(X \leq 6) - \text{P}(X \leq 2)$ .

Probability and cumulative probability built-in functions can be graphed when one of $n$ , $p$ , $x$ is unknown, and can also be tabulated.

Example: A coin is manufactured to land on head 52% of the time and the rest on its tail.

a) 800 people are each flipping such a coin 10 times in a row. Find the probability that exactly one person gets all tails or all heads.

b) Find the number of people needed to have a 95% chance of at least two people each flipping at least eight tails or at least eight heads in ten flips.

a) Let $X$ be the number of heads in ten flips, such that ${X \sim \text{B}(10, 0.52)}$ .

\begin{align*} \text{success} &= \text{P}(X = 0) + \text{P}(X = 10) \\ &= \binom{10}{0}0.52^0 \cdot 0.48^{10} + \binom{10}{10}0.52^{10} \cdot 0.48^0 \\ &\approx 0.0020948 \dots \end{align*}

Then let $Y$ be number of times someone gets all tails or heads in ten flips, out of $800$ people. ${Y \sim \text{B}(800, 0.0020948)}$ .

\begin{align*} \text{P}(Y = 1) &= \binom{800}{1}0.0020948^1 \cdot (1 - 0.0020948)^{799} \\ &\approx 0.314 \qed \end{align*}

b) At least 8 tails or at least 8 heads means

\text{P}(X \leq 2) + \text{P}(X \geq 8)

using the $X$ from previous part.

Many calculators can only do cumulative binomial distribution probability from 0 to $x$ , so this is

\text{P}(X \leq 2) + 1 - \text{P}(X \leq 7) = 0.11218749\dots

Now we can define ${Z \sim \text{B}(n, 0.11218749)}$ for number of people getting at least 8 tails or at least 8 heads and solve

\text{P} (Z \geq 2) > 0.95

or equivalently

\text{P}(Z \leq 1) < 0.05

As $n$ is a discrete value, taking on only positive integers, this cannot be solved via solver, but rather through a table of values.

$n$	$\text{P}(Z \leq 1)$
$39$	$0.0572$
$40$	$0.0519$
$41$	$0.0470$
$42$	$0.0426$

As $n = 41$ is the first value to drop below $0.0500$ , the answer is $n = 41 \qed$

Expected value

The expected value is

\text{E}(X) = np

This is why

\text{expected count} = \text{number of trials}\times\text{probability}

Variance and standard deviation

\text{Var}(X) = np(1-p)

\sigma = \sqrt{np(1-p)}

Some ways that they could be used on exam is if you are given expected value and variance and you need to solve for $n$ and $p$ .

Mode and median

The mode and median are very close to $np$ . Precise formulas are beyond scope of the course. If asked on exam, use methods discussed in discrete random variables to investigate.

The median is not necessarily equal to the mode(s).

Tips

Does it say at most ( $\leq N$ ), at least ( $\geq N$ ), less than( $< N$ ), more than ( $> N$ ), or equal to ( $=N$ )?
See also probability formulas.
See also binomial expansion.