Introduction to Geometric Distributions (College Board AP® Statistics)

Conditions for geometric distributions

What is a geometric distribution?

• A discrete random variable follows a geometric distribution if it counts the number of trials needed to obtain the first success when an experiment satisfies the following conditions:

  • the trials are independent of each other

  • there are exactly two outcomes of each trial (success or failure)

  • the probability of success () is constant for each trial

• The notation is sometimes used for a geometric random variable

  • is the probability of success

    • is the probability of failure

• can take any positive integer 1, 2, 3, ...

  • cannot be 0 as it counts the number of trials until the first success

Modeling with geometric distributions

What can be modeled by geometric distributions?

• A geometric distribution can model any variable that satisfies the three conditions

  • e.g. the number of times a coin is flipped until it lands on a heads for the first time

    • Each coin flip is independent of each other

    • A success is landing on heads and a failure is landing on tails

    • The probability of landing on heads is constant as the same coin is used each time

• Geometric models can sometimes still be used when it seems like there are more than two outcomes

  • e.g. the number of times a six-sided dice is rolled until it lands on a six for the first time

    • Although the dice could land on one of six numbers, for this experiment there are only two outcomes: landing on a six (success) or not landing on a six (failure)

  • Geometric distributions are good models in these situations provided that a successful outcome can be clearly defined

What cannot be modeled by geometric distributions?

• Geometric models are not suitable if:

  • the trials are not independent

    • e.g. the number of tokens taken from a bag containing 6 blue tokens and 4 red tokens until a blue token is first obtained

  • there are more than two outcomes

    • e.g. the number of emails received in an hour

  • the probability of success changes

    • e.g. the number of attempts an archer takes until they first hit a target if they take a step closer to the target after each shot

Properties of geometric distributions

What is the shape of a geometric distribution?

• The mode of any geometric distribution is 1

• The heights of the bars in the diagram are always decreasing

• All geometric distributions are skewed to the right

How do I calculate the mean and standard deviation of a geometric distribution?

• If then

  • the mean is

  • the standard deviation is

    • These formulas are given in the exam