The Binomial Distribution | CIE AS Maths: Probability & Statistics 1 Revision Notes 2020

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Properties of Binomial Distribution

What is a binomial distribution?

A binomial distribution is a discrete probability distribution
The discrete random variable X follows a binomial distribution if it counts the number of successes when an experiment satisfies the conditions:
- There are a fixed finite number of trials $(n)$
- The outcome of each trial is independent of the outcomes of the other trials
- There are exactly two outcomes of each trial (success or failure)
- The probability of success (p) is constant
If X follows a binomial distribution then it is denoted $X ~ B (n, p)$
- $n$ is the number of trials
- $p$ is the probability of success
The probability of failure is 1-p which is sometimes denoted as q
The formula for the probability of r successful trials is given by:
- $P = (X = r) = (\begin{matrix} n \\ r \end{matrix}) p^{r} {(1 - p)}^{n - r}$ for r = 0, 1, 2,....,n
- This is equal to the term which includes $p^{r}$ in the expansion of ${(p + q)}^{n}$ where $q = 1 - p$ (this shows the link with the Binomial Expansion)

What are the important properties of a binomial distribution?

The expected number (mean) of successful trials is $n p$
The variance of the number of successful trials is $n p (1 - p)$
- Square root to get the standard deviation
The distribution can be represented visually using a vertical line graph
- If p is close to 0 then the graph has a tail to the right
- If p is close to 1 then the graph has a tail to the left
- If p is close to 0.5 then the graph is roughly symmetrical
- If p = 0.5 then the graph is symmetrical

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Modelling with Binomial Distribution

How do I set up a binomial model?

Identify what a trial is in the scenario
- For example: rolling a dice, flipping a coin, checking hair colour
Identify what the successful outcome is in the scenario
- For example: rolling a 6, landing on tails, having black hair
Make sure you clearly state what your random variable is
- For example, let X be the number of students in a class of 30 with black hair

What can be modelled using a binomial distribution?

Anything that satisfies the four conditions
For example, let $T$ be the number of times a fair coin lands on tails when flipped 20 times: $T ~ B (20, \frac{1}{2})$
- A trial is flipping a coin: There are 20 trials so n =20
- We can assume each coin flip does not affect subsequent coin flips: They are independent
- A success is when the coin lands on tails: Two outcomes - tails or not tails (heads)
- The coin is fair: The probability of tails is constant with $p = \frac{1}{2}$
Sometimes it might seem like there are more than two outcomes
- For example, let Y be the number of yellow cars that are in a car park full of 100 cars
- Although there are more than two possible colours of cars, here the trial is whether a car is yellow so there are two outcomes (yellow or not yellow)
- Y would still need to fulfil the other conditions in order to follow a binomial distribution
Sometimes a sample may be taken from a population
- For example, 30% of people in a city have blue eyes, a sample of 30 people from the city is taken and X is the number of them with blue eyes
- As long as the population is large and the sample is random then it can be assumed that each person has a 30% chance of having blue eyes

What can not be modelled using a binomial distribution?

Anything where the number of trials is not fixed or is infinite
- The number of emails received in an hour
- The number of times a coin is flipped until it lands on heads
Anything where the outcome of one trial affects the outcome of the other trials
- The number of caramels that a person eats when they eat 5 sweets from a bag containing 6 caramels and 4 marshmallows
  - If you eat a caramel for your first sweet then there are less caramels left in the bag when you choose your second sweet
Anything where there are more than two outcomes of a trial
- A person's shoe size
- The number a dice lands on when rolled
Anything where the probability of success changes
- The number of times that a person can swim a length of a swimming pool in under a minute when swimming 50 lengths
  - The probability of swimming a lap in under a minute will decrease as the person gets tired

Worked example

It is known that 8% of a large population are immune to a particular virus. Mark takes a sample of 50 people from this population. Mark uses a binomial model for the number of people in his sample that are immune to the virus

(a)

State the distribution that Mark uses.

(b)

State the two assumptions that Mark must make in order to use a binomial model.

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Examiner Tip

If you are asked to criticise a binomial model always consider whether the trials are independent, this is usually the one that stops a variable from following a binomial distribution!

The Binomial Distribution (CIE AS Maths: Probability & Statistics 1)

Revision Note

Properties of Binomial Distribution

What is a binomial distribution?

What are the important properties of a binomial distribution?

Modelling with Binomial Distribution

How do I set up a binomial model?

What can be modelled using a binomial distribution?

What can not be modelled using a binomial distribution?

Worked example

Examiner Tip

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Author: Amber