What is a geometric distribution and its notation?

• A?geometric?distribution?is a?discrete probability distribution
• 
  • 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

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
    • ? ? ? ? ? ? ? ? ? ? ? ?(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

What are the properties of a 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”

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)
• The?mode?is?1?for all geometric distributions