10.04 Applications of general discrete random variables

 

When we are asked to model a particular situation using a discrete probability distribution, we are being asked to read and interpret an experiment and represent the probability distribution as a table OR a function.

In this section where we're mainly dealing with 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 the 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 (such as combinations)

Step Four: Calculate your probabilities and represent them in a table together with each outcome.

 

Worked examples

Example 1

A fair standard die 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.

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 die. 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.

Example 2

A pencil case contains $5$5 blue pens and $3$3 red pens. $4$4 pens are drawn randomly from the pencil case without replacement.

Let $X$X represent the number of blue pens drawn.

a)  Describe the probability distribution for $X$X.

Think: Describing a probability distribution is the same as constructing a probability distribution. This particular situation will involve the use of combinations and counting techniques.

Do: We'll construct a table of values. Note we'll need at least $1$1 blue pen if we want a total of $4$4 pens.

$x$x	$1$1	$2$2	$3$3	$4$4
$P(X=x)$P(X=x)	$\frac{\nCr{5}{1}\nCr{3}{3}}{\nCr{8}{4}}$5C13C38C4​	$\frac{\nCr{5}{2}\nCr{3}{2}}{\nCr{8}{4}}$5C23C28C4​	$\frac{\nCr{5}{3}\nCr{3}{1}}{\nCr{8}{4}}$5C33C18C4​	$\frac{\nCr{5}{4}\nCr{3}{0}}{\nCr{8}{4}}$5C43C08C4​

Now we'll simplify our table to give us these probabilities.

$x$x	$1$1	$2$2	$3$3	$4$4
$P(X=x)$P(X=x)	$\frac{1}{14}$114​	$\frac{6}{14}$614​	$\frac{6}{14}$614​	$\frac{1}{14}$114​

b)  Calculate the probability of between $1$1 and $4$4 blue pens being drawn from the pencil case.

Think: We can use our table to add all the relevant probabilities. Note that between $1$1 and $4$4 does not include $1$1 nor $4$4.

Do: $P(X=2)+P(X=3)=\frac{6}{14}+\frac{6}{14}=\frac{12}{14}$P(X=2)+P(X=3)=614​+614​=1214​

c) Calculate $P(X>1|X<4)$P(X>1|X<4)

Think: In words, this question is asking what is the probability of more than $1$1 blue pen being drawn if we know less that less than $4$4 blue pens were drawn from the pencil case. We can use our result from part b).

Do: 

$P(X>1|X<4)$P(X>1|X<4)	$=$=	$\frac{P(X>1\cap X<4)}{P(X<4)}$P(X>1∩X<4)P(X<4)​
	$=$=	$\frac{P\left(X=2\right)+P\left(X=3\right)}{P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right)}$P(X=2)+P(X=3)P(X=1)+P(X=2)+P(X=3)​
	$=$=	$\frac{\frac{12}{14}}{\frac{13}{14}}$1214​1314​​
	$=$=	$\frac{12}{13}$1213​

 

Practice questions

Question 1

Question 2

Question 3