7. Discrete Random Variables

7.02 Discrete random probability distribution

7.03 Measures of centre and spread of discrete random variables

7.04 Applications of discrete random variables

Lesson

Worksheet

Practice

7.05 Pascal's triangle (Investigation)

7.06 Pascal's triangle

7.07 Binomial distributions

7.08 Graphs of binomial distribution using a graphics calculator (Investigation)

7.09 Problems with binomial distributions using calculator

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7.04 Applications of discrete random variables

Lesson

Worksheet

Practice

Worksheet

Applications

An investment scheme advertises the following percentage returns after $2$ years based on historical probabilities:

$\text{Percentage return } (x\text{)}$	$10\%$	$15\%$	$25\%$
$P \left(X = x \right)$	$0.7$	$0.15$	$0.15$

Find the expected percentage return on an investment.

Find how much an investment of $\$50\,000$ is expected to be worth after $2$ years.

A salesperson is starting work in a new region and analyses the probability of how many sales he is likely to make in the next month:

$\text{Number of sales } (x)$	$0$	$1$	$2$	$3$	$4$
$P(X = x)$	$0.35$	$0.25$	$0.2$	$0.15$	$0.05$

Given that he makes at least one sale, state the probability that he will make $2$ sales.

The salesperson is offered $2$ payment schemes:

Option A: Flat monthly income of $\$1800$
Option B: $\$1000$ flat fee per month plus $\$500$ per sale

If he chooses option B, what is his expected monthly income?

Which option should he choose to maximise his income?

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

Cars with just a driver pay $\$25$
Cars with a driver and one passenger pay $\$15$
Cars with a driver and at least two passengers pay $\$12$

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

Number of people	$1$	$2$	$3$	$4$	$5$
Number of cars	$4500$	$3500$	$1100$	$600$	$300$

Find the probability a randomly selected car is carrying $3$ people.

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

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

Find the expected revenue per car in this car park.

Find the standard deviation of the revenue per car in this car park.

To be allowed to leave class and go to lunch, Mrs Ammar gives her class a $3$ -question multiple choice quiz, with each question consisting of $4$ possible answers. David hasn’t listened to a word Mrs Ammar has said all lesson and will have to guess each of the three questions.

Find the probability David will guess none of the questions correct.

Find the probability David will guess all of the questions correctly.

Find the probability David will guess just one of the questions correctly.

Let $X$ represent the number of questions David guesses correctly. Construct a probability distribution table for $X$ .

State whether the table represents a discrete probability distribution. Explain your answer.

When a student completes a task set by their teacher on Spacemaths, the number of hints used is monitored by the system:

The probability of using at least $1$ hint is $0.6$ .
The probability of using $2$ hints is the same as using $3$ hints.
The probability of using $1$ hint is the same as using $4$ hints.
At most students can use $4$ hints.
The probability they use $2$ hints is half the probability that they use $0$ hints.

Let $X$ represent the number of hints they used. Construct a probability distribution table for $X$ .

Find the expected number of hints a student will use.

Given that a student used at least $2$ hints, find the probability they used $4$ hints.

Jo and Ky are playing a game of cards that is either won or lost, there is no draw. The probability that Jo wins the first game is $0.6$ .

If Jo wins a game, the probability he wins the next game is $0.7$ .
If Jo loses a game, the probability that Ky wins the next game is $0.8$ .
They keep playing until either Jo or Ky wins two games.

Construct a probability tree diagram of this situation.

Let $X$ represent the number of games of cards played before someone wins two games. Construct a probability distribution table for $X$ .

Find $E \left(X\right)$ .

Given that Jo won, calculate the probability that $3$ games were played.

A fair standard die is thrown. Let $X$ be the number of dots on the uppermost face.

Construct the probability distribution table for $X$ .

Is the discrete probability distribution uniform or non-uniform?

Find $E \left(X\right)$ .

Find $\text{Var} \left(X\right)$ .

Two normal dice are rolled and the sum of the numbers on the uppermost face recorded. Let $Y$ represent the value of the sum of the two dice.

Complete the table for this discrete probability distribution:

$y$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$P ( Y = y )$	$\dfrac{1}{36}$		$\dfrac{3}{36}$			$\dfrac{6}{36}$					$\dfrac{1}{36}$

State the most likely sum to occur.

Describe the shape of the distribution.

Find the expected value.

Find the variance.

A die was manufactured such that an odd number is twice as likely to be rolled as an even number. Let $X$ be the number the die lands on.

Construct a probability distribution table for $X$ .

Find $E \left(X\right)$ .

Find $\text{Var} \left(X\right)$ .

A game is played in which a standard six-sided die is rolled. If it lands on a number other than $1$ , then the score is that number. If it lands on $1$ , then a second four-sided dice with numbers $3$ to $6$ is rolled and the number that die lands on is the score. Let $X$ be the score of a player in this game.

Construct a probability distribution table for $X$ .

Find $E \left( X \right)$ .

Find $\text{Var} \left(X\right)$ .

A six-sided die with numbers from $1$ to $6$ is weighted such that $P \left(\text{ prime number }\right) = 0.1$ and $P \left( 4 \right) = P\left(6\right) = 0.3$ . Let $X$ represent the possible outcomes from one roll of the dice.

Construct the probability distribution table for $X$ .

Find the following:

P ( X < 3 )

P ( X = 3 | X \leq 5 )

iii

P ( X < 3 | X < 5 )

P ( X < 4 | X \geq 2)

A fair standard die is thrown onto the ground and the number of visible odd-numbered faces (the faces which are not on the ground) is noted. Let $Y$ be the number of visible odd-numbered faces.

Construct the probability distribution for $Y$ .

Is this discrete probability distribution uniform or non-uniform?

A fair standard die is rolled and the number of dots on the visible faces (the faces which are not on the ground) is noted. Let $W$ be the number of dots that can be seen on the visible faces.

Construct the probability distribution table for $W$ .

Is the discrete probability distribution uniform or non-uniform?

A regular six-sided dice has a side length of $8\text{ cm}$ . The dice is rolled on the ground and the height above ground of the dot on the face with only a single dot is noted. Let $H$ be the number of centimetres this single dot is above the ground.

List the possible outcomes for $H$ .

Hence, construct the probability distribution table for $X$ .

Two dice are rolled and the absolute value of the differences between the numbers appearing uppermost are recorded.

Complete the sample space in the given table.

Let $X$ be defined as the absolute value of the difference between the two dice. Construct the probability distribution table for $X$ .

Find $P ( X < 3 )$

Find $P ( X \leq 4 | X \geq 2)$

	$1$	$3$	$4$	$5$
$1$	$0$		$3$
$2$	$1$
$3$				$2$
$4$
$5$	$4$	$2$
$6$

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

Complete the sample space in the given table.

Let $X$ be the winnings from one game. Construct a probability distribution table for $X$ .

Find the expected winnings.

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

	$1$	$2$	$3$
$1$	$0$	$1$	$2$
$2$
$3$
$4$
$5$
$6$

At a local fair, in a game that involves rolling a standard six-sided die, players can win a prize depending on what they roll. Each player must pay $\$3$ to play. The prizes are awarded as follows:

The player wins $\$3$ if a $1$ , $3$ or $5$ is rolled.
The player wins $\$6$ if a $4$ or $6$ is rolled.
The player wins $\$9$ if a $2$ is rolled.

Let $X$ be the prize received by the player. Construct a probability distribution table for $X$ .

Find the expected prize value.

Find the standard deviation of the distribution.

At a fair, a games stall operator offers prizes worth $\$1.50$ , $\$2$ , $\$1$ and $\$0.50$ for one attempt at a particular game. The probabilities of winning these prizes are respectively $0.15$ , $0.01$ , $0.05$ and $0.03$ .

Find the probability of not winning a prize.

If each game costs $\$2$ , find the expected profit per game for the operator in dollars.

If each game costs $C$ and the games stall operator made a profit of $\$595$ from $500$ games. Find $C$ , the amount he likely charged per game in dollars.

The probability that a particular biased coin lands on tails is $0.7$ . Let $X$ be the number of tails when the coin is tossed twice. Complete the given probability distribution table for $X$ .

$x$	$0$	$1$	$2$
$P(X = x)$

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.

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

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

An unfair coin is tossed. The chance of tails facing upwards after the toss is $30\%$ .

Find the probability of the coin landing tails up for the first time on the third toss.

Find the probability of the coin landing tails up for the first time on the fourth toss.

Find the probability that it takes four tosses of the coin before you see a tail on the fifth toss.

Let $N$ be the number of tosses of the coin it takes before you see a tail on the next toss. Define the probability density function for $N$ .

Two fair spinners, A and B, are spun. The number from each spinner is noted and the total score is defined below:

X = \begin{cases} A + B; \text{ if } A = B \\ A + B; \text{ if } A\gt B \\ B - A; \text{ if } A\lt B\end{cases}

List all the possible outcomes of $X$ .

Construct a probability distribution for $X.$

A spinner has four sections each numbered $1$ to $4$ . The spinner is divided according to the given equations. Let $X$ represent the number spun on the spinner.

$P \left( 1 \right) = P \left( 2 \right) + P \left( 3 \right) + P \left( 4 \right)$
$P \left( 2 \right) = 2 P \left( 3 \right)$
$P \left( 3 \right) = P \left( 4 \right)$

Construct the probability distribution table for $X$ .

Hence, construct the cumulative probability distribution table for $X$ .

Find the following:

$P(X < 3)$ .

$P(X \geq 3)$ .

iii

$P(X=1 \cup X=3)$ .

$P(X \leq 3|X > 1)$ .

Two spinners numbered from $0$ to $4$ are spun. Let $X$ be the product of the two numbers that come up.

List all the possible outcomes of $X$ .

Construct a probability distribution table for $X$ .

Find $E \left( X \right)$ .

Find $\text{Var} \left(X\right)$ .

Three marbles are randomly drawn from a bag containing seven black and three green marbles. Let $X$ be the number of black marbles drawn.

Construct the probability distribution table for $X$ if the marbles are drawn with replacement.

Construct the probability distribution table for $X$ if the marbles are drawn without replacement.

In Brad’s toy box, there are $3$ toy cars and $4$ toy dinosaurs. Each day, for three days, he takes a toy at random and plays with it, and then puts it back.

Construct a probability tree diagram of all the possible combination of toys he could have played with over these three days.

Let $X$ be the number of days he played with a toy car. Construct the probability distribution table for the discrete random variable $X$ .

Find the expected number of days he will play with the toy car.

Find the standard deviation for the distribution of $X$ .

A pencil case contains $6$ blue pens and $5$ green pens. $4$ pens are drawn randomly from the pencil case without replacement.

Find the probability of drawing one blue pen from the pencil case.

Find the probability of drawing three blue pens from the pencil case.

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

$x$	$0$	$1$	$2$	$3$	$4$
$P \left(X = x \right)$	$\dfrac{1}{66}$		$\dfrac{5}{11}$

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

Find the probability of drawing one blue pen from the pencil case.

Find the probability of drawing three blue pens from the pencil case.

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

$x$	$0$	$1$	$2$	$3$	$4$
$P \left(X = x \right)$	$\dfrac{625}{38\,416}$		$\dfrac{6075}{19\,208}$

Two earrings are taken without replacement from a draw containing $3$ black earrings and $5$ brown earrings. Let $X$ be the number of black earrings drawn.

Construct a probability distribution for $X$ .

Given that at least one black earring was selected, find the probability that two were selected.

A child randomly selects $3$ balls without replacement from a box containing $8$ red balls and $2$ green balls. For every red ball chosen, the child receives $1$ chocolate. For every green ball chosen, the child receives $5$ chocolates.

Let $X$ be the number of green balls chosen. Construct the probability distribution table for $X$ .

Let $T$ be the number of chocolates the child receives. Construct the probability distribution table for $T$ .

State the most likely number of chocolates that the child receives.

State the expected number of chocolates that the child receives.

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7.04 Applications of discrete random variables