# Discrete Probability Distributions

## What is a discrete probability distribution?

A Discrete Probability Distribution tells you the various probabilities associated with a discrete random variable. What does this mean? The definition of the word `distribution’ refers to how something is shared out in a group or how it is spread out over an area. Let X be a random variable – a variable whose possible values are the outcomes of a random event (see more on Random Variables). If X is a discrete random variable then its possible values belong to a discrete set of outcomes. If there are three possible outcomes then:

$X\in\left\lbrace x_1,x_2,x_3\right\rbrace$.

For example, a spinner is numbered 1, 2 and 3 with probabilities of 0.2, 0.35 and 0.45 respectively. This is an example of a discrete random variable and its associated discrete probability distribution.

## The Uniform Distribution

The uniform probability distribution describes a discrete distribution where each outcome has an equal probability. Consider a random variable X that has a discrete uniform distribution. X can take one of k values:

$X\in\left\lbrace x_1,x_2,x_3,…,x_k\right\rbrace$.

If all these values all equally likely then they must each have a probability of 1/k. This is because they must all add up to 1. See Example 1.

## The Binomial Distribution

The binomial probability distribution is closely related to Binomial Expansion. It gives the probability distribution of a random variable X that is subject to a number of trials. Each trial is independent (probabilities do not change from trial to trial) has a probability of success or failure. For example, the number of times a head comes up when a coin is tossed repeatedly is a binomial random variable. Each coin toss is a trial and probability of success is the probability of tossing heads.

In general, suppose there are n trials and each trial has a probability of success p. We write $X\sim B(n,p)$ where $\sim$ means ‘distribution’ and ‘B’ means Binomial. The probability of having r successes is $P(X=r)$ and given by:

$P(X=r)=\left(\begin{array}{c}n\\r\end{array}\right)p^r(1-p)^{n-r}$.

Look up Binomial Expansion to calculate the ‘n choose r’ term in brackets. Now see Example 2. We can use the tables in the Edexcel Formula Booklet (pages 29-33) to find cumulative probabilities. This is where you are looking for the probability of r successes or fewer: $P(X\leq r)$. Look for your value of n down the left hand side of each box in the table. Look for your value of p across the top. You will then see the r (or x) values on the left of each box for which you can read off the corresponding probability inside the table. See Example 3.

## Discrete Probability Distribution Examples

Consider a single dice roll. The set of possible outcomes is {1,2,3,4,5,6} – this is discrete. Each of these outcomes has a probability of 1/6 and therefore represents a discrete uniform distribution.

A coin is tossed 10 times. The number of heads is a binomial random variable. X may take integer between 0 and 10 inclusively. Each of the outcomes has a different probability. To find the probability of tossing 4 heads specifically we put n=10, p=0.5 and r=4 into the formula:

$\left(\begin{array}{c}10\\4\end{array}\right)(0.5)^4(0.5)^6=\frac{105}{512}\approx 0.2051.$

This is because p and 1-p are both 0.5, i.e. the probability of success (tossing heads is success) and failure (tossing tails is failure) are both the same. The number of trials is n=10 which is the number of tosses. The probability we are looking for is obtaining r=4 successes.

Suppose $X\sim B(20,0.25)$ and we wish to find $P(6<X\leq 10)$. Out of a possible 20 trials, we are looking for the probability of getting 7, 8, 9 OR 10 successes. The Excel Formula booklet with n=20, p=0.25 and x=10 gives the cumulative probability of getting less than or equal to 10 successes as $P(X\leq 10)=0.9961$. We can also read off the probability of getting less than or equal to 6 successes as $P(X\leq 6)=0.7858$. The probability of getting 7, 8, 9 or 10 can be found from:

$P(6<X\leq 10)=P(X\leq 10)-P(X\leq 6)$

Hence,

$P(6<X\leq 10)=0.2103$.