The Binomial Distribution (AQA AS Maths): Revision Note

Properties of Binomial Distribution

What is a binomial distribution?

• A binomial distribution is a discrete probability distribution

• The discrete random variable 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 

  • 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 

  • is the number of trials

  • 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:

  • 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)

  • You will be expected to use the distribution function on your calculator to calculate probabilities with the binomial distribution

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 p = 0.5 then the graph is symmetrical

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 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 

• 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   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