The Geometric Probability Distribution
Hi there. In this math/probability post, I go over the geometric probability distribution. This may be basic but this particular distribution seems to be overlooked.
A Discrete Probability Distribution
A geometric random variable X
is considered discrete as it is for modeling the first success after a repeated number of similar trials. The number of trials are integers (whole numbers) which makes this random variable discrete. The probability of success is denoted by p
and the probability of failure is 1-p
.
The geometric probability distribution is a discrete probability distribution where the trials keep occurring until the first success. We keep "failing" and try until the first success.
Denote F
for fail and S
for success. You could have:
FFFFFFFS
where the first success is after seven failures
or
FS
with the first success after the second try.
or
FFFFFFFFFFS
where the first success happens after ten tries.
Note that the probability of failure is the same in each trial.
The Geometric Distribution Formula
The probability of the x-th success
for a geometric probability distribution is given by:
for x = 1, 2, 3, 4, ...
You could try to memorize this formula but I suggest an alternative memory aid. The probability of the x-th trial being a success is given by the probabilities of the failures being (1 - p)
before the x-th trial. At the x-th trial you obtain the first success symbolized by the probability of success p
.
Mean and Variance Of A Geometric Random Variable
For the geometric random variable we have the mean (expected value) and the variance for it.
The mean is given by .
The variance here is.
Derivations for the mean and variance is a bit involved. It involves Taylor Series (in which I have forgotten). For those interested you can look at this resource.
Cumulative Distribution Function
For the geometric distribution, the probability distribution formula is for computing the probability of the x-th
success for x = 1, 2, 3... .
What if you want the probability of the first success happening within a certain amount of trials? We have the cumulative distribution function for the probability of the first success occurring in x
amount of trials or less.
where p
is the probability of success in a trial.
Examples
Example One
There is a spinner with four different colours. The colours are red, green, blue and yellow. Every colour has an equal chance of being landed on (0.25 each). What is the probability of landing on blue for the first time on the third spin?
For this question compute the probability when x = 3.
There is a 14.0625% chance of landing blue for the first time on the third spin.
Example Two
Assume a fair die with numbers from 1 to 6 (inclusive). You roll a single die multiple times until a three is rolled. What is the probability of landing a first three on the seventh roll?
There is about a 5.58% chance of landing a first three on the seventh roll.
Example Three
Use the same spinner from question one. What is the probability of landing on yellow for the first time in four spins?
You could do long way of computing the probabilities of a first yellow on the first spin, one for the second spin, third spin and fourth spin. The faster way here would be using the cumulative distribution function for computer the probability of a yellow for the first time in four spins or fewer.
There is about a 68.36% chance of landing on a yellow for the first time in four spins.
Example Four
Use the same die from question two. What is the expected number of rolls for rolling a three for the first time?
The keyword expected should remind you of expected values and means. The probability of rolling a three on a die in a trial is one-sixth. This is the value for the probability of success.
Computing the expected number of rolls for rolling a three involved the mean formula for the geometric random variable.
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