# Discrete probability distributions

Discrete probability distributions relate to distributions for discrete variables. As described in Random variables, **discrete variables are distinct and countable variables**, like number of persons, number of visitors, prices on goods, etc.

## Discrete probability distributions example:

*How many times over the next 100 times that I go to the post office at 2 pm can I expect to have a line with 3 persons in front of me?*

This question refers to the number of persons. The number cannot be a number of persons with a long tail of decimals. Persons are not divisible. So, **this question relates to a discrete probability distribution.**

Based on observations that I would have carried out; I can construct the probability distribution which will help giving an estimate of how many times **I can expect there to be 3 persons** in line over my next 100 visits in the post office at 2 pm.

## Web shop example

Let’s expand on the post-office-example from above:

**Say that I am selling goods through a web shop** and that I do the shipment via my local post office and that I go there at 2 pm some 100 times a year. Therefore, I am interested in knowing how many people are in line (although, what really interests me, is the amount of time I must wait and not the number of people, but for the sake of the exercise…).

Say I have done 60 observations, so far:

Based on my 60 observations in the post office,** there is a 0.13 probability that I will find 3 persons in line**. So, I would estimate that over the next 100 visits, for 0.13 x 100 = 13 of these visits there will 3 persons in line.

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## Coin flipping and probability distributions

*What is the probability that we get one tail when flipping two fair coins at a time?* The number of tails that we can get can be defined as a random variable. We would say: “Let X be the number of tails we flip” and to answer the question we would calculate for *“What is the probability of X=1?”*

There are four total possible outcomes for flipping two coins at a time:

- Head, Head
- Head, Tail
- Tail, Tail
- Tail, Head

The probability for each of these outcomes has a 1/4. But we are being asked about the probability of 1 tail to occur.

There are two events that we have one tail, {Head, Tail} and {Tail, Head}, so this gives two fourths = one half or 0.5 probability. We would express the probabilities of the four possible outcomes of the random variables in this way:

By listing the probabilities for each of the possible outcomes of the random variables, **we have constructed its’ probability distribution** which can be visualized through a histogram:

## Examples of discrete probability distributions

In my material, I describe and apply the following discrete probability distributions:

- The Bernoulli distribution
- The binomial distribution
- The Poisson distribution
- The geometric distribution
- The hypergeometric distribution

Learning resources

- Discrete probability distributions on Jbstatistics Youtube playlist
- Jbstatistics video: An Introduction to Discrete Random Variables and Discrete Probability Distributions

#### Carsten Grube

Freelance Data Analyst

##### Normal distribution

##### Two-sample inference

##### Confidence intervals

##### Simple linear regression, fundamentals

##### ANOVA & the F-distribution

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