The binomial distribution

​​In a nutshell

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.

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.

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.

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The probability of $$x$$ successful outcomes is given by​

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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

$$X \backsim B(10,0.3)$$

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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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