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7.04 Applications of general discrete random variables

Worksheet
Applications
1

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)0.70.150.15
a

Find the expected percentage return on an investment.

b

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

2

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)01234
P(X = x)0.350.250.2 0.150.05
a

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

b

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?

c

Which option should he choose to maximise his income?

3

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 people12345
Number of cars450035001100600300
a

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

b

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

c

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

d

Find the expected revenue per car in this car park.

e

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

4

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.

a

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

b

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

c

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

d

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

e

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

5

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.

a

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

b

Find the expected number of hints a student will use.

c

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

6

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.

a

Construct a probability tree diagram of this situation.

b

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

c

Find E \left(X\right).

d

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

7

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

a

Construct the probability distribution table for X.

b

Is the discrete probability distribution uniform or non-uniform?

c

Find E \left(X\right).

d

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

8

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.

a

Complete the table for this discrete probability distribution:

y23456789101112
P ( Y = y )\dfrac{1}{36}\dfrac{3}{36}\dfrac{6}{36}\dfrac{1}{36}
b

State the most likely sum to occur.

c

Describe the shape of the distribution.

d

Find the expected value.

e

Find the variance.

9

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.

a

Construct a probability distribution table for X.

b

Find E \left(X\right).

c

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

10

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.

a

Construct a probability distribution table for X.

b

Find E \left( X \right).

c

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

11

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.

a

Construct the probability distribution table for X.

b

Find the following:

i
P ( X < 3 )
ii
P ( X = 3 | X \leq 5 )
iii
P ( X < 3 | X < 5 )
iv
P ( X < 4 | X \geq 2)
12

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.

a

Construct the probability distribution for Y.

b

Is this discrete probability distribution uniform or non-uniform?

13

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.

a

Construct the probability distribution table for W.

b

Is the discrete probability distribution uniform or non-uniform?

14

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.

a

List the possible outcomes for H.

b

Hence, construct the probability distribution table for X.

15

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

a

Complete the sample space in the given table.

b

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

c

Find P ( X < 3 )

d

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

123456
103
21
32
4
542
6
16

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.

a

Complete the sample space in the given table.

b

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

c

Find the expected winnings.

d

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

123456
1012
2
3
4
5
6
17

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.

a

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

b

Find the expected prize value.

c

Find the standard deviation of the distribution.

18

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.

a

Find the probability of not winning a prize.

b

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

c

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.

19

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.

x012
P(X = x)
20

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.

a

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

b

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.

21

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

a

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

b

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

c

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

d

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.

22

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}
a

List all the possible outcomes of X.

b

Construct a probability distribution for X.

23

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)

a

Construct the probability distribution table for X.

b

Hence, construct the cumulative probability distribution table for X.

c

Find the following:

i

P(X < 3).

ii

P(X \geq 3).

iii

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

iv

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

24

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

a

List all the possible outcomes of X.

b

Construct a probability distribution table for X.

c

Find E \left( X \right).

d

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

25

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

a

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

b

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

26

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.

a

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

b

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

c

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

d

Find the standard deviation for the distribution of X.

27

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

a

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

b

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

c

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

x01234
P \left(X = x \right)\dfrac{1}{66}\dfrac{5}{11}
28

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.

a

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

b

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

c

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

x01234
P \left(X = x \right)\dfrac{625}{38\,416}\dfrac{6075}{19\,208}
29

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.

a

Construct a probability distribution for X.

b

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

30

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.

a

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

b

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

c

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

d

State the expected number of chocolates that the child receives.

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Outcomes

ACMMM137

use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable

ACMMM142

use discrete random variables and associated probabilities to solve practical problems

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