The binomial distribution is a model which can be used to calculate probabilities of success from a statistical experiment which involves repeated trials. Each trial has two outcomes under the model: success or failure. The binomial distribution can only be used under certain conditions. Binomial probabilities can be calculated using a formula or using the distribution function on a calculator.

Conditions for a binomial distribution

The binomial distribution can be used if these four conditions are satisfied.

$1.$	There are a fixed number of trials, $n$ .
$2.$	There are two possible outcomes, success or failure.
$3.$	The probability of success, $p$ , is constant.
$4.$	Trials are independent of each other.

Calculating binomial probabilities

Calculate probabilities using the formula

Consider a statistical experiment with two trials, with the probability of success as $p$ . The tree diagram represents the outcomes and probabilities.

Maths; Binomial distribution; KS5 Year 12; The binomial distribution

The probability of two successful outcomes is given by $p^2$ , whereas the probability of one success is given by $2 \times p \times (1-p)$ as there are two possible ways of getting one successful outcome, $p$ , and one failure, $1-p$ .

With more trials, rather than using a tree diagram to show the probabilities, it is easier to use a formula to calculate binomial probabilities for $n$ trials, with probability of success $p$ . The binomial distribution is defined in the following way.

$X \backsim B(n,p)$

$X$	A random variable which follows a binomial distribution.
$B$	The binomial distribution.
$n$	The total number of trials.
$p$	The probability of success.

The probability of $x$ successful outcomes is given by

$P(X=x)={^n}C_x\space p^x \space (1-p)^{(n-x)}$

$n$	The total number of trials.
$p$	The probability of success.
$x$	The total number of successes.
${^n}C_x$	The number of different ways or combinations of selecting $x$ from $n$ .

Example 1

A biased coin is tossed $10$ times. The probability of obtaining a head is $0.3$ . Find the probability of obtaining $4$ heads.

Let $X$ be the number of heads, which can be modelled by the binomial distribution as

$X \backsim B(10,0.3)$

To calculate the probability of obtaining $4$ heads, let $x=4$ and use the formula.

$\begin {aligned}P(X=4)& = {^{10}}C_4 \times 0.3^4 \times 0.7^6 \\&=0.200 \space (3 s.f.)\end {aligned}$

The probability of obtaining $4$ heads is $\underline{0.200}$ to $(3 \ s.f.)$ 

Find probabilities using the distribution function on a calculator

Binomial probabilities can be found from a calculator using the distribution function. Select the binomial probability distribution (Bpd) and enter $x, N$ and $p$ .

CALCULATOR TIP

$\boxed{\begin{aligned}Binomial \space &pd \\Data &: variable \\x&: \\N &: \\p&: \\\end {aligned}}$

Example 2

The probability that a biased die lands on one is $0.2$ . The die is rolled $20$ times. Find the probability that the die lands on one $6$ times.

Let $X$ be the number of times the die lands on one.

$X \backsim B(20,0.2)$ 

To calculate $P(X=6)$ , use the calculator with $x=6$ , $N=20$ and $p=0.2$ .

	$\boxed{\begin{aligned}Binomial \space &pd \\Data &: variable \\x&: 6 \\N &: 20\\p&: 0.2\\\end {aligned}}$
	$\boxed{p = \qquad \qquad \quad 0.109}$

Therefore, $\underline{P(X=6) = 0.109 \space (3 \space s.f.)}$

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