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AustraliaVIC
VCE 12 Methods 2023

8.04 Applications of general discrete random variables

Lesson

 

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>1X<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}}$12141314
  $=$= $\frac{12}{13}$1213

 

Practice questions

Question 1

At a car park in the city, all day parking is charged on the following basis:

  • Cars with just a driver pay $\$20$$20
  • Cars with a driver and one passenger pay $\$18$$18
  • Cars with a driver and at least two passengers pay $\$10$$10

The number of people in one of these cars on a given day is summarised in the table.

Number of people $1$1 $2$2 $3$3 $4$4 $5$5
Number of cars $4600$4600 $3400$3400 $1100$1100 $600$600 $300$300
  1. Calculate the probability a randomly selected car is carrying $3$3 people.

  2. Given that a car was carrying at least $2$2 people, what is the probability it was carrying $4$4?

  3. Let $X$X represent the parking fee paid by a randomly selected car. Construct the probability distribution for $X$X below.

    Write the possible values of $X$X in descending order from left to right.

    $x$x $\editable{}$ $\editable{}$ $\editable{}$
    $P$P$($($X=x$X=x$)$) $\editable{}$ $\editable{}$ $\editable{}$
  4. Calculate the expected revenue per car in this car park.

Question 2

In a game of two-up, a person called the “Spinner” tosses two coins.

If the coins land with two heads up, then the Spinner wins and the gamblers lose.

If the coins land with two tails up, the Spinner loses and the gamblers win.

If the coins land one head up and one tail up, the Spinner tosses the coins again and the gamblers break even.

  1. Construct a tree diagram to represent all possible outcomes of tossing two coins.

  2. If each gambler bets $\$3$$3, and can win $\$3$$3 per toss, use the table below to construct the probability distribution for the profit of the gambler for one game of two-up.

    Enter the possible outcomes from left to right in ascending order. Leave out the currency symbol.

    $x$x $\editable{}$ $\editable{}$ $\editable{}$
    $P$P$($($X=x$X=x$)$) $\editable{}$ $\editable{}$ $\editable{}$

Question 3

Two dice are rolled and the difference between the largest number and smallest number is calculated. A player wins $\$1$$1 if the difference is $3$3, $\$2$$2 if the difference is $4$4, $\$3$$3 if the difference is $5$5 and $\$0$$0 otherwise.

  1. Complete the following array, which shows the difference between the dice rolls.

      $1$1 $2$2 $3$3 $4$4 $5$5 $6$6
    $1$1 $0$0 $1$1 $2$2 $\editable{}$ $\editable{}$ $\editable{}$
    $2$2 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
    $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
    $4$4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
    $5$5 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
    $6$6 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Let $X$X be the winnings from one game. Complete the probability distribution table for $X$X.

    $x$x $\$0$$0 $\$1$$1 $\$2$$2 $\$3$$3
    $P\left(X=x\right)$P(X=x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  3. Find the expected winnings in dollars.

  4. If it costs $\$2$$2 to play each game, find the player's expected return in dollars.

Outcomes

U34.AoS4.2

discrete random variables: - specification of probability distributions for discrete random variables using graphs, tables and probability mass functions - calculation and interpretation of mean, 𝜇, variance, 𝜎^2, and standard deviation of a discrete random variable and their use - Bernoulli trials and the binomial distribution, Bi(𝑛, 𝑝), as an example of a probability distribution for a discrete random variable - effect of variation in the value(s) of defining parameters on the graph of a given probability mass function for a discrete random variable - calculation of probabilities for specific values of a random variable and intervals defined in terms of a random variable, including conditional probability

U34.AoS4.10

calculate and interpret the probabilities of various events associated with a given probability distribution, by hand in cases where simple arithmetic computations can be carried out

U34.AoS4.11

apply probability distributions to modelling and solving related problems

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