3.2.3 The Geometric Distribution

What is a geometric distribution and its notation?

• A geometric distribution is a discrete probability distribution
• The discrete random variable follows a geometric distribution if it counts the number of trials until the first success occurs for an experiment that satisfies the conditions
  • Each trial has only two outcomes
    Broadly labelled as “success” and “failure” - these are mutually exclusive

Such trials may be referred to as Bernoulli trials – named after the Swiss mathematician Jacob Bernoulli (1655-1705)

• • The outcomes of trials are independent
    The outcome of one trial does not affect the outcome of another trial
  • The probability of each outcome is constant across all trials

i.e.   The probability of “success” does not change between trials

• p is the probability of “success” in a single trial
  • The probability of “failure” in a single trial is 1- p ; often denoted by q
• If X  follows a geometric distribution, it is denoted by 
• The formula for finding the probability that the first success occurs after r trials (or after r -1 failures) is

• e.g. If the probability of success in a single trial is 0.3, then the probability it will take 6 trials to obtain the first success is given by

What does a geometric distribution look like?

• If represented visually, using a vertical line graph, the probabilities in a geometric distribution decrease but never reach zero
  • The probabilities form a geometric progression
    • Question: What would the first term (“a”), the common ratio (“r”) and the sum to infinity be (“”)?
                               (Answer below diagram)
    • The probabilities decrease exponentially; drawing a curve through the tops of the lines would produce a decreasing exponential curve
  • The graphs also show how 1 is the mode for every geometric distribution

• Answer:

What are the properties of a geometric distribution?

• for all r
  • Every geometric distribution has an infinite (discrete) sample space which is the set of natural numbers () or positive integers ()
• for all r
  • The mode of every geometric distribution is 1
    (the value of r that has the highest probability)
• Geometric distributions are memoryless 
  • The number of trials needed for the first success is not dependent on the number of trials that have already occurred
    e.g.   If 5 (failed) trials have already occurred, the probability of the first success happening after 7 trials is the same as the probability of it happening after 2 trials which would be 
  • Mathematically this is written as

 
where is the number of trials that have already occurred

e.g.      

• Geometric distributions have the recurrence relation 
• The mean, or expected value, of a geometric distribution is 

How do I set up a geometric model?

• Identify what a trial is in the context of a problem
  • Flipping a coin, rolling a dice, a football match
• Identify a successful outcome
  • Heads, a square number, a win
• Identify the probability of “success”
  • 
• Define your random variable using the correct notation
  • Let X be the number of trials required to obtain the first heads when flipping a fair coin,  

What can be modelled using a geometric distribution?

• Anything where the first occurrence of a successful outcome is of significance
  • Rolling a double with two dice before being allowed to start a game
  • The number of on/off presses a switch can withstand before wearing out

(In which case the first “success” would be the first failure of the switch!)

• In addition, the scenario must satisfy the three conditions
  • Trials only have two outcomes of interest
  • Trials are independent
  • Probability of success is constant for all trials
  • These are also three of the four conditions for a binomial distribution so are not enough on their own – it will also depend on the context
• Many scenarios may appear as having more than two outcomes but in the context of the question only two are of significance
  • e.g. A light that randomly flashes in 8 different colours, but the only colour of interest is blue
             So “blue” is “success” and all other colours, regardless of whether it  is red, yellow, etc – i.e. “not blue” - is “failure”
• Sometimes a sample may be taken from a population
  • As long as the population is large enough and the sample is random the probability of “success” in the sample is the same as the probability of “success” in the population

What cannot be modelled using a geometric distribution?

• Be careful not to confuse binomial and geometric distributions/models
  • Binomial is for the number of successes in a fixed number of trials
  • Geometric is for the number of trials up to and including the first success
• Anything where a trial would have more than two outcomes of interest
  • e.g. Outcome of a football match – win, draw or lose
• Where the probability of an outcome of a trial is influenced by a previous trial
  • i.e. trials are not independent
  • e.g. drawing counters from a bag without replacement
• Anything where the probability of “success” changes with time – or practice
  • e.g. a skateboarder performing a trick - the probability of success should increase after practising the trick

How do I calculate geometric probabilities?

• Identify p, the probability of “success” and 1- p, the probability of “failure” (q)
• For exact probabilities use 
• For inequalities use
  • 
    • This means the first success occurs after r trials, therefore the first  trials all ended in failure
    • Similarly, 
  • 
    • Similarly, 
  • 
    • If a and b are close it may be easier to use

• • Logic can be used to deduce most geometric distribution questions so memorising these formulae is not essential
• Beware of questions that exploit the memorylessness property of geometric distributions – loosely called “given that” questions
  • e.g.   means
            “the probability that  equals 8 given that  is greater than 6” or
             “the probability of 8 trials given that 6 trials have already occurred”
                                   
• The mean (expected value) or mode of a geometric distribution may be required
  •  
  • The mode is 1 for all geometric distributions