The Geometric Distribution (CIE A Level Maths: Probability & Statistics 1)

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Properties of Geometric Distribution

What is a geometric distribution and its notation?

  • A geometric distribution is a discrete probability distribution
  • The discrete random variable X 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 X tilde Geo left parenthesis p right parenthesis
  • The formula for finding the probability that the first success occurs after r trials (or after r -1 failures) is

begin mathsize 16px style bold italic P bold left parenthesis bold italic X bold equals bold italic r bold right parenthesis bold equals bold left parenthesis bold 1 bold minus bold p bold right parenthesis to the power of bold r bold minus bold 1 end exponent bold italic p end style

  • 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

P left parenthesis X equals 6 right parenthesis equals left parenthesis 1 minus 0.3 right parenthesis to the power of 6 minus 1 end exponent left parenthesis 0.3 right parenthesis equals left parenthesis 0.7 right parenthesis to the power of 5 open parentheses 0.3 close parentheses equals 0.050421

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 (“S subscript infinity”)?
                                 (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

3-2-3-cie-fig0-geo-dist-graphs

  • Answer: a equals p comma r equals q equals 1 minus p comma space S subscript infinity equals 1

What are the properties of a geometric distribution?

  • straight P left parenthesis X equals r right parenthesis greater than 0 for all r
    • Every geometric distribution has an infinite (discrete) sample space which is the set of natural numbers (straight natural numbers) or positive integers (straight integer numbers to the power of plus)
  • straight P open parentheses X equals r close parentheses less than straight P left parenthesis X equals r minus 1 right parenthesis 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 begin mathsize 16px style open parentheses 1 minus p close parentheses p end style
    • Mathematically this is written as

straight P left parenthesis X equals r vertical line X greater than k right parenthesis equals straight P left parenthesis X equals r minus k right parenthesis 
where k is the number of trials that have already occurred

e.g.   bold P bold left parenthesis bold italic X bold equals bold 7 bold vertical line bold italic X bold greater than bold 5 bold right parenthesis equals straight P left parenthesis X equals 7 minus 5 right parenthesis equals bold italic P bold left parenthesis bold italic X bold equals bold 2 bold right parenthesis   

  • Geometric distributions have the recurrence relation P left parenthesis X equals r right parenthesis equals q P left parenthesis X equals r minus 1 right parenthesis
  • The mean, or expected value, of a geometric distribution is bold italic E bold left parenthesis bold italic X bold right parenthesis bold equals bold 1 over bold p

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Modelling with Geometric Distribution

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
    • begin mathsize 16px style 0.5 comma 1 third comma 42 percent sign end style
  • 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,  begin mathsize 16px style X tilde Geo left parenthesis 0.5 right parenthesis end style

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 “successchanges with time – or practice
    • e.g. a skateboarder performing a trick - the probability of success should increase after practising the trick

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Calculating Geometric Probabilities

How do I calculate geometric probabilities?

  • Identify p, the probability of “success” and 1- p, the probability of “failure” (q)
  • For exact probabilities use straight P left parenthesis X equals r right parenthesis equals left parenthesis 1 minus p right parenthesis to the power of r minus 1 end exponent p
  • For inequalities use
    • straight P left parenthesis X greater than r right parenthesis equals left parenthesis 1 minus p right parenthesis to the power of r
      • This means the first success occurs after r trials, therefore the first r  trials all ended in failure
      • Similarly, straight P left parenthesis X greater or equal than r right parenthesis equals straight P left parenthesis X greater than r minus 1 right parenthesis equals q to the power of r minus 1 end exponent
    • straight P left parenthesis X less than r right parenthesis equals 1 minus straight P left parenthesis X greater than r minus 1 right parenthesis equals 1 minus q to the power of r minus 1 end exponent
      • Similarly, straight P left parenthesis X less or equal than r right parenthesis equals 1 minus straight P left parenthesis X greater than r right parenthesis equals 1 minus q to the power of r
    • straight P left parenthesis a less or equal than X less or equal than b right parenthesis equals straight P left parenthesis X less or equal than b right parenthesis minus straight P left parenthesis X less than a right parenthesis
      • If a and b are close it may be easier to use

begin mathsize 16px style P left parenthesis X equals a right parenthesis plus P left parenthesis X equals a plus 1 right parenthesis plus... plus P left parenthesis X equals b right parenthesis end style
 

    • 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.  P left parenthesis X equals 8 vertical line X greater than 6 right parenthesis 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”
                    P left parenthesis X equals 8 vertical line X greater than 6 right parenthesis equals P left parenthesis X equals 8 minus 6 right parenthesis equals P left parenthesis X equals 2 right parenthesis                 
  • The mean (expected value) or mode of a geometric distribution may be required
    •  bold E bold left parenthesis bold italic X bold right parenthesis bold equals bold 1 over bold italic p
    • The mode is 1 for all geometric distributions

Worked example

Given that X tilde Geo left parenthesis 0.4 right parenthesis find

(i)
straight P left parenthesis X equals 5 right parenthesis
(ii)
straight P left parenthesis X greater than 5 right parenthesis
(iii)
straight P left parenthesis 3 less or equal than X less than 8 right parenthesis
(iv)
straight P left parenthesis X equals 8 space vertical line space X greater than 5 right parenthesis
(v)
The mode of  X.

3-2-3-cie-fig1-we-solution-1

Examiner Tip

  • Try not to get bogged down with formulae for the geometric distribution, most questions can be deduced using logic
  • If you are asked to criticise a geometric model always consider whether trials are independent
    • especially if it involves “practising” or “performing” a skill
    • most people will improve after they’ve made several attempts at a skill
    • so the probability of success should gradually increase over time
  • If finding the number of trials required (r) then be careful counting calculator presses; remember you are likely to be finding r minus 1(the number of failures before success) in the first instance

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.