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ASMaths: StatisticsAQARevision Notes4. Statistical Distributions4.2 Binomial Distribution4.2.1 The Binomial Distribution

The Binomial Distribution (AQA AS Maths: Statistics)

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Dan

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Dan

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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 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 begin mathsize 16px style bold left parenthesis bold n bold right parenthesis end style
    • 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 begin mathsize 16px style X tilde B left parenthesis n comma p right parenthesis end style
    • begin mathsize 16px style n end style is the number of trials
    • begin mathsize 16px style p end style 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:
    • begin mathsize 16px style P equals left parenthesis X equals r right parenthesis equals open parentheses table row n row r end table close parentheses space p to the power of r space left parenthesis 1 minus p right parenthesis to the power of n minus r end exponent end style for r = 0, 1, 2,....,n
    • This is equal to the term which includes begin mathsize 16px style p to the power of r end style in the expansion of begin mathsize 16px style left parenthesis p plus q right parenthesis to the power of n end style where begin mathsize 16px style q equals 1 minus p end style (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 begin mathsize 16px style bold italic n bold italic p end style
  • The variance of the number of successful trials is begin mathsize 16px style bold italic n bold italic p bold left parenthesis bold 1 bold minus bold italic p bold right parenthesis end style 
    • 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

 

4-2-1-the-binomial-distribution-diagram-1-part-1

4-2-1-the-binomial-distribution-diagram-1-part-2

4-2-1-the-binomial-distribution-diagram-1-part-3-1

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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: begin mathsize 16px style T tilde straight B open parentheses 20 comma 1 half close parentheses end style
    • 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 equals 1 half
  • 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.
(a)
State the distribution that Mark uses.

4-2-1-the-binomial-distribution-we-solution-part-1

(b)
State the two assumptions that Mark must make in order to use a binomial model.
4-2-1-the-binomial-distribution-we-solution-part-2

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!

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.