### Properties of Binomial Distribution

#### What is a binomial distribution?

- A binomial distribution is a
**discrete probability distribution** - The
**discrete random variable**follows a*X***binomial distribution**if it**counts the number of successes**when an experiment satisfies the conditions:- There are a
**fixed finite number of trials** - 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**

- There are a
- If
*X*follows a binomial distribution then it is denoted- is the number of trials
- is the probability of success

- The
**probability of failure is 1-**which is sometimes denoted as*p***q** - The formula for the probability of
*r*successful trials is given by:- for
*r =*0, 1, 2,..*..,n* - This is equal to the term which includes in the expansion of where (this shows the link with the Binomial Expansion)

- for

#### What are the important properties of a binomial distribution?

- The
**expected number (mean)**of successful trials is - The
**variance**of the number of successful trials is- 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**then the graph is*p =*0.5**symmetrical**

- If

### 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

- For example, let

#### What can be modelled using a binomial distribution?

- Anything that satisfies the
**four conditions** - For example, let be the number of times a fair coin lands on tails when flipped 20 times:
- 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

- A trial is flipping a coin: There are 20 trials so
- 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

- For example, let
- 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

- For example, 30% of people in a city have blue eyes, a sample of 30 people from the city is taken and

#### 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

- The number of caramels that a person eats when they eat 5 sweets from a bag containing 6 caramels and 4 marshmallows
- 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

- The number of times that a person can swim a length of a swimming pool in under a minute when swimming 50 lengths

#### Exam 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!