New Zealand
Level 8 - NCEA Level 3

# Constructing Discrete Probability Distribution

Lesson

When we are required to construct our own discrete probability distribution, there are a number of things that this could mean.

• Read and interpret an experiment and represent the probability distribution as a table
• Read and interpret an experiment and represent the probability distribution as a function

In this section where we're mainly dealing with general discrete random variables, we'll mostly be constructing tables to represent our distributions.

## How to construct a discrete probability distribution

Step One: Define exactly what your random variable $X$X represents.

Note: You can use any capital letter for your random variable, but most often we use $X$X

Step Two: Consider how many possible outcomes $X$X can take.

Step Three: Devise a method for calculating the probabilities for each outcome.

Note: The main methods you will use here is either constructing a sample space (the most common being a tree diagram or a table) or using counting techniques (combinations)

Step Four: Calculate your probabilities and represent them in a table.

You have now created an individual probability distribution for your random variable!

#### Worked Examples

##### Example 1:

A fair standard dice is thrown and the number of dots on the uppermost face is noted.

Let $X$X be the number of dots on the uppermost face.

(a)  Construct the probability distribution for $X$X.

Think: In this example the random variable has already been defined for us. All we need to do is determine the number of possible outcomes and then work out their associated probabilities.

Do: The outcomes for $X$X are $1,2,3,4,5$1,2,3,4,5 and $6$6 since we're talking about rolling a standard dice. We also know the probabilities for each are $\frac{1}{6}$16 since each outcome is equally likely.

Now we construct a table of values.

$x$x $1$1 $2$2 $3$3 $4$4 $5$5 $6$6
$P(X=x)$P(X=x) $\frac{1}{6}$16 $\frac{1}{6}$16 $\frac{1}{6}$16 $\frac{1}{6}$16 $\frac{1}{6}$16 $\frac{1}{6}$16

And there we have our probability distribution.

(b)  State the type of distribution which $X$X represents.

Think: Our discrete distributions can either be uniform or non - uniform. Later on you'll learn about specific types of non-uniform discrete random variables.

Do: Since all outcomes have the same probability, this distribution is a uniform distribution.

##### QUESTION 1

A random variable $X$X can take any of the values $0$0, $1$1, $2$2 or $3$3. The following facts about the distribution of $X$X are known.

• $P\left(X=0\right)=\frac{1}{2}P\left(X=1\right)$P(X=0)=12P(X=1)
• $P\left(X=1\right)=4P\left(X=2\right)$P(X=1)=4P(X=2)
• $P\left(X=2\right)=\frac{1}{5}P\left(X=3\right)$P(X=2)=15P(X=3)
1. Use these facts to complete the probability distribution table for $X$X. Give each probability in simplest form.

 $x$x $P\left(X=x\right)$P(X=x) $0$0 $1$1 $2$2 $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

##### QUESTION 2

A pencil case contains $9$9 blue pens and $5$5 green pens. $4$4 pens are drawn randomly from the pencil case, one at a time, each being replaced before the next one is drawn.

1. What is the probability of drawing one blue pen from the pencil case?

2. What is the probability of drawing three blue pen from the pencil case?

3. Let $X$X be the number of blue pens drawn. Complete the probability distribution table.

 $x$x $P\left(X=x\right)$P(X=x) $0$0 $1$1 $2$2 $3$3 $4$4 $\frac{625}{38416}$62538416​ $\editable{}$ $\frac{6075}{19208}$607519208​ $\editable{}$ $\editable{}$

### Outcomes

#### S8-4

Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal

#### 91586

Apply probability distributions in solving problems