Poisson & Geometric Probability
Probability&Statistics
Study Guides
Big Picture Distributions
The geometric and Poisson probability distributions are two other discrete probability distributions that are related
to the binomial distribution. Both of these probability distributions can be applied to solve real-world problems. The
geometric probability distribution is useful for determining the number of trials needed to achieve success, while the
Poisson distribution is useful for describing the number of events that will occur during a specific interval of time or in
a specific distance, area, or volume.
Key Terms
Geometric Distribution: The discrete probability distribution of the number of trials needed to achieve success.
Poisson Distribution: The discrete probability distribution of the number of events that occur in a specific time
interval or space.
Geometric Probability Distribution
Experiment Geometric Distribution
• Experiment consists of a sequence of independent • The geometric distribution is found by calculating
trials the geometric probabilities for n = 1, 2, 3, ....
• Each trial has two outcomes: success or failure • A discrete probability distribution because n can only
• The number of trials is not fixed; instead, the be whole numbers
experiment continues until the first success • As n increases, P(x = n) decreases
• The probability of success is the same for each
Mean for the geometric distribution:
trial.
The experiment is essentially binomial trials repeated Standard deviation for the binomial distribution:
until the first success is achieved, and then the
experiment stops
• Example: the number of times a coin needs to be
Figure: Geometric probability distributions for three different val-
tossed until the first head (success) appears ues of p.
• Useful in business applications–example: how
many candidates need to be interviewed before
the perfect candidate for a job is found?
Probability
• Random variable X is the number of trials until the
first success appears
• To calculate the probability of getting a success on
the nth trial, P(x = n) = (1-p)n-1(p), where n is a
whole number and p is the probability of success
(this value is the same for each trial)
• To directly find the probability of more than n
trials completed before there is one success,
you would need to sum the probabilities of
an infinite number of trials, which would be
impossible. Instead, use the complement rule:
P(x > n) = 1 - P(x ≤ n)
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Poisson & Geometric Probability
Poisson Probability Distribution
Experiment
Distributions
Poisson Distribution
cont.
• Experiment consists of counting the number of • The Poisson distribution is found by calculating the
events that will occur during a specific interval Poisson probabilities for n = 1, 2, 3, ....
of time or in a specific distance, area, or volume
• A discrete probability distribution because n can only be
• There are two outcomes: the event occurs whole numbers
(success) or does not occur (failure)
• Each event is independent Mean for the geometric distribution: μ = λ
Probability
• The probability that an event occurs during the Standard deviation for the binomial distribution:
specified time interval or space is the same
The experiment is a special case where the number
of binomial trials gets larger and the probability of
success gets smaller
• Example: the number of traffic accidents at a
particular intersection
• Useful in predicting or estimating a number of
things – planes at an airport, the number of
fishes caught by a fisherman, arrival times, etc.
Probability
• Random variable X is the number of events that
occur (successes)
• To calculate the probability of n events,
,
where λ is the mean number of events in the
time, distance, volume, or area
Figure: Poisson probability distributions for three different values of
• e is approximately equal to 2.7183
λ.
• To directly find the probability of more than
events occuring, you would need to sum the For a binomial distribution where the number of trials n ≥ 100
and the probability of success p where np < 100, then the
probabilities of an infinite number of trials, binomial distribution for k successes can be approximated
which would be impossible. Instead, use the with a Poisson distribution where λ = np
complement rule.
Graphing Calculator
In a graphing calculator, we can use built-in commands to find the geometric and Poisson distributions.
Geometric Distribution
The command for geometric distribution is: geometpdf(p, x). p is the probability of success, and x is the trial that we
want the success to occur in. This will give us the probability of success occurring on that trial.
There is another similar equation called geometcdf, which requires us to plug in two values for x: one low and one high.
It will give us the probability of success occurring between those two trials.
Poisson Distribution
The command for Poisson distribution is: poissonpdf(λ, x). λ is the expected number of events, and x is the number of
events. This will give us the probability that x many events occurred.
There is a similar command called poissoncdf, which requires us to plug in two values for x: one low and one high. This
will give us the probability the number of events that occurred fell between these two numbers.
If you can’t find these commands, check the manual for your graphing calculator. For the TI-83/TI-84, both commands
are found by pressing [2ND][DISTR].
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