4.5.1 The Binomial Distribution

What is a binomial distribution? 

• A binomial distribution is a discrete probability distribution
• A discrete random variable  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 follows a binomial distribution then it is denoted 
  • 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:
  • for 
    • where 
  • 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

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

• • 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

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 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 p = 0.5
• Sometimes it might seem like there are more than two outcomes
  • For example: let 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)
    • 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
      • The probability is not constant