DP IB Maths: AI HL

Revision Notes

4.7.1 The Binomial Distribution

Test Yourself

Properties of Binomial Distribution

What is a binomial distribution? 

  • A binomial distribution is a discrete probability distribution
  • A discrete random variable X follows a binomial distribution if it counts the number of successes when an experiment satisfies the following 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 is constant (p)
  • If X follows a binomial distribution then it is denoted X tilde straight B left parenthesis n comma space p right parenthesis
    • 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:
    • straight P left parenthesis X equals r right parenthesis equals C presuperscript n subscript r cross times p to the power of r left parenthesis 1 minus p right parenthesis to the power of n minus r end exponent for r equals 0 comma space 1 comma space 2 comma space... comma space n
      • C presuperscript n subscript r equals fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction where n factorial equals n cross times left parenthesis n minus 1 right parenthesis cross times left parenthesis n minus 2 right parenthesis cross times... cross times 3 cross times 2 cross times 1
    • You will be expected to use the distribution function on your GDC to calculate probabilities with the binomial distribution

What are the important properties of a binomial distribution? 

  • The expected number (mean) of successful trials is

straight E left parenthesis X right parenthesis equals n p 

    • You are given this in the formula booklet
  • The variance of the number of successful trials is

Var left parenthesis X right parenthesis equals n p left parenthesis 1 minus p right parenthesis 

    • You are given this in the formula booklet
    • 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

1Aka39bs_4-2-1-the-binomial-distribution-diagram-1-part-1shQh2uVX_4-2-1-the-binomial-distribution-diagram-1-part-2

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

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
  • Identify the parameters
    • n is the number of trials and p is the probability of success in each trial
  • 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:
    • A trial is flipping a coin: There are 20 trials so = 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 = 0.5
  • 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
        • The probability is not constant

Exam Tip

  • An exam question might involve different types of distributions so make it clear which distribution is being used for each variable

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.

4-5-1-ib-ai-aa-sl-modelling-binomial-a-we-solution

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

4-5-1-ib-ai-aa-sl-modelling-binomial-b-we-solution

c)
Calculated the expected number of people in the sample that are immune to the virus.

4-5-1-ib-ai-aa-sl-modelling-binomial-c-we-solution

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Dan

Author: Dan

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.