Topic 2: Discrete random variables

Check the probabilities sum to 1, then E(X) = sum of x p(x); for a binomial, name n and p and check the trials are independent with a fixed p.

A lesson for this topic

Four-option practice

Question 1Practice

Probability distribution of X: P(X = 0) = 0.2, P(X = 1) = 0.3, P(X = 2) = k, P(X = 3) = 0.15.
x	0	1	2	3
P(X = x)	0.2	0.3	k	0.15

The discrete random variable

X

has the probability distribution shown in the table, where

k

is a constant. What is the value of

k

Question 2Practice

Probability distribution of X: P(X = 1) = 0.25, P(X = 2) = 0.3, P(X = 3) = 0.35, P(X = 4) = 0.1.
x	1	2	3	4
P(X = x)	0.25	0.3	0.35	0.1

The number of pets,

X

, owned by a randomly chosen family in a survey of an Adelaide suburb has the probability distribution shown in the table. What is

E(X)

Question 3Practice

The number of breakdowns,

X

, of a school bus in a week has the probability distribution

P(X

= 0)

= 0.3

P(X

= 1)

= 0.5

and

P(X

= 2)

= 0.2

. What is the standard deviation of

X

Question 4Practice

A random variable

X

has mean 5 and standard deviation 4. Let

Y

= 3X - 4

. What is the standard deviation of

Y

Question 5Practice

A random variable

X

has

\text{Var}(X)

= 1.5

. What is

\text{Var}(4X + 2)

Question 6Practice

At a school fair, a game costs $2.50 to play. A player spins a spinner and wins $8 with probability

\frac{1}{8}

, $3 with probability

\frac{1}{4}

, and nothing otherwise. Let

G

be the player's net gain in dollars for one game (the prize minus the $2.50 paid). What is

E(G)

Question 7Practice

A survey of 200 nests of a seabird on the Eyre Peninsula recorded the number of eggs in each nest. The bar chart shows the results. Based on these results, what is the best estimate of the probability that a randomly chosen nest has at least 2 eggs?

Question 8Practice

A spinner has five equal sectors numbered 2, 4, 6, 8 and 10. Let

X

be the number it lands on. What is the standard deviation of

X

Question 9Practice

A discrete random variable

X

has

P(X

= x)

= kx

for

x

= 1, 2, 3, 4

, where

k

is a constant. What is

P(X

\ge 3)

Question 10Practice

Probability distribution of X: P(X = 0) = a, P(X = 1) = 0.35, P(X = 2) = b, P(X = 3) = 0.1.
x	0	1	2	3
P(X = x)	a	0.35	b	0.1

The random variable

X

has the probability distribution shown in the table, where

a

and

b

are constants, and

E(X)

= 1.45

. What is the value of

a

Question 11Practice

The number of customers,

X

, who arrive at a bakery in Hahndorf in a 5-minute period has

E(X)

= 2.4

. Which statement is correct?

Question 12Practice

The random variable

X

takes the values 1, 2 and 3 with probabilities 0.1, 0.8 and 0.1. The random variable

Y

takes the values 1, 2 and 3 with probabilities 0.4, 0.2 and 0.4. Which statement is correct?

Question 13Practice

In a card game at a school fundraiser, a player wins $20 with probability

\frac{1}{10}

, wins $5 with probability

\frac{1}{5}

, and wins nothing otherwise. The organisers want to make a mean profit of 40 cents per game. What should they charge to play?

Question 14Practice

A random variable

X

has

E(X)

= 3

and

E(X^2)

= 13

. What is the standard deviation of

X

Question 15Practice

A random variable

X

has

E(X)

= 2

and

\text{Var}(X)

= 1.5

. What is

E(X^2)

Question 16Practice

Two fair six-sided dice are rolled. Let

X

be the larger of the two numbers shown (if both dice show the same number,

X

is that number). What is

P(X

= 4)

Question 17Practice

Probability distribution of X: P(X = 0) = 0.05, P(X = 1) = 0.15, P(X = 2) = 0.3, P(X = 3) = 0.25, P(X = 4) = 0.175, P(X = 5) = 0.075.
x	0	1	2	3	4	5
P(X = x)	0.05	0.15	0.3	0.25	0.175	0.075

The number of goals,

X

, scored by a school hockey team in a match has the probability distribution shown in the table. Let

\mu = E(X)

and

\sigma = \text{SD}(X)

. What is

P(\mu - \sigma

< X

< \mu + \sigma)

Question 18Practice

Probability distribution of X: P(X = 1) = 0.35, P(X = 2) = 0.2, P(X = 3) = 0.25, P(X = 4) = 0.2.
x	1	2	3	4
P(X = x)	0.35	0.2	0.25	0.2

The number of rounds,

X

, that a golfer at a club in the Barossa Valley plays in a week has the probability distribution shown in the table. What is the standard deviation of

X

, correct to three significant figures?

Question 19Practice

A random variable

X

takes the values 1, 2 and 3. Which list could give

P(X

= 1)

P(X

= 2)

and

P(X

= 3)

for a valid probability distribution?

Question 20Practice

A spinner is spun 50 times and lands on red 18 times. Which statement is correct?

Question 21Practice

Each tomato seedling at a nursery in the Adelaide Hills is diseased with probability 0.36. Let

Y

= 1

if a randomly chosen seedling is diseased and

Y

= 0

otherwise. What is the standard deviation of

Y

Question 22Practice

A Bernoulli random variable has variance 0.21, and its probability of success is less than 0.5. What is its probability of success?

Question 23Practice

Which random variable has a Bernoulli distribution?

Question 24Practice

For a Bernoulli random variable with probability of success

p

, which value of

p

gives the largest standard deviation, and what is that standard deviation?

Question 25Practice

A Bernoulli random variable

Y

has standard deviation 0.4, and a success is more likely than a failure. What is

E(Y)

Question 26Practice

X

has a binomial distribution with

n

= 4

and

p

= \frac{1}{3}

. What is

P(X

= 2)

Question 27Practice

X

has a binomial distribution with

n

= 50

and

p

= 0.2

. What is the standard deviation of

X

Question 28Practice

Each customer at an ice-cream shop in Glenelg chooses a waffle cone with probability 0.35, independently of other customers. What is the probability that at least 5 of the next 12 customers choose a waffle cone, correct to three significant figures?

Question 29Practice

Each seedling planted in a revegetation project along the River Murray survives its first summer with probability 0.6, independently of the others. Twenty seedlings are planted. What is the probability that between 10 and 14 seedlings, inclusive, survive their first summer, correct to three significant figures?

Question 30Practice

Each ticket in a school raffle wins a prize with probability 0.1, independently of other tickets. What is the smallest number of tickets a person must buy for the probability of at least one winning ticket to be more than 0.95?

Question 31Practice

X

has a binomial distribution with mean 6 and standard deviation 2. What is the number of trials,

n

Question 32Practice

Which random variable has a binomial distribution?

Question 33Practice

X

has a binomial distribution with

n

= 4

and

P(X

= 0)

= 0.0081

. What is

E(X)

Question 34Practice

An orchard on the Eyre Peninsula plants 15 almond seeds. Each seed germinates with probability 0.82, independently of the others. What is the probability that at most 11 seeds germinate, correct to three significant figures?

Question 35Practice

Which binomial distribution is closest to symmetric and bell-shaped?

Question 36Practice

X

has a binomial distribution with

n

= 5

and

p

= 0.1

. What is

P(X

\ge 2)

Question 37Practice

A nursery sows 40 seeds, which cost $20 in total. Each seed grows into a seedling with probability 0.8, independently, and each seedling sells for $3. The profit is

3X - 20

dollars, where

X

is the number of seedlings. What is the standard deviation of the profit, in dollars, correct to three significant figures?

Question 38Practice

A quiz has 20 questions, each with four options of which one is correct. A student guesses every answer at random. What is the probability that the student gets at least 8 questions correct, correct to three significant figures?

Question 39Practice

Five cards are drawn one at a time, without replacement, from a standard deck of 52 playing cards. Let

X

be the number of hearts drawn. Why does

X

not have a binomial distribution?

Question 40Practice

X

has a binomial distribution with

n

trials and probability of success

p

. If

n

is multiplied by 4 and

p

is unchanged, what happens to the standard deviation of

X

Question 41Practice

X

has a binomial distribution with

n

= 5

, and

P(X

= 1)

= P(X

= 2)

. What is

\text{Var}(X)

Question 42Practice

A discrete random variable

X

has

P(X

= x)

= k(x + 1)

for

x

= 0, 1, 2, 3

, where

k

is a constant. What is

E(X)

Question 43Practice

In a survey of an Adelaide suburb, each household chosen at random has a rainwater tank with probability 0.3, independently of the others. Of 25 households chosen at random, what is the probability that at most 6 have a rainwater tank, correct to three significant figures?

Question 44Practice

Probability distribution of X: P(X = 1) = 0.1, P(X = 2) = 0.3, P(X = 3) = a, P(X = 4) = b.
x	1	2	3	4
P(X = x)	0.1	0.3	a	b

The random variable

X

has the probability distribution shown in the table, where

a

and

b

are constants. The random variable

Y

= 10 - 2X

has

E(Y)

= 4.6

. What is the standard deviation of

Y

Question 45Practice

Each scratch card sold at a school fundraiser wins a prize with probability 0.05, independently of the other cards. What is the least number of cards a person must buy for the probability of winning at least two prizes to be more than 0.9?

Question 46Practice

A courier makes 8 deliveries a day in the Barossa Valley. Each delivery is late with the same probability, independently of the others, and the expected number of late deliveries in a day is 2. On a day when at least one delivery is late, what is the probability that at most two deliveries are late, correct to three significant figures?

Question 47Practice

At a school fair, a player spins a fair spinner 18 times. The spinner has three equal sectors, one of them red. Each spin that lands on red wins the player $4, and each spin that does not land on red loses the player $1. What are the mean and the standard deviation of the player's total profit?

Question 48Practice

Each batch of solar panels made at a factory in Adelaide has the same number of panels, and each panel passes a quality test with the same probability, independently of the others. The number of panels in a batch that pass has a binomial distribution with mean 20 and standard deviation 2. What is the probability that fewer than 3 panels in a batch fail the test, correct to three significant figures?

Question 49Practice

A carton of 12 eggs holds 3 cracked eggs. A cook takes 4 eggs from the carton at random, one at a time, without putting any back. Let

X

be the number of cracked eggs taken. A student models

X

as binomial with

n

= 4

and

p

= \frac{1}{4}

. Is the student's model correct, and what is

P(X

= 1)

Short response questions

Tokens from a spinnerShort response · 9 marks

At a school fair, a spinner gives a number of tokens,

X

. The probability distribution of

X

is shown in the table, where

k

is a constant.

Probability distribution of X: P(X = 0) = 0.4, P(X = 1) = k, P(X = 2) = 2k, P(X = 5) = 0.15.
x	0	1	2	5
P(X = x)	0.4	k	2k	0.15

(a) Find the value of

k

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Find

E(X)

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

\text{Var}(X)

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Find

P(X

> E(X))

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Interpret the value of

E(X)

in the context of the spinner. [2]

0 words · a sentence or two

A marble game at a school fairShort response · 9 marks

At a school fair, a bag holds 4 red and 2 blue marbles. A player pays $3 and then draws two marbles at random without replacement. Two blue marbles win $10, one marble of each colour wins $2, and two red marbles win nothing. Let

G

be the player's net gain in dollars (the prize minus the $3 paid).

(a) Find the probability that the player draws one marble of each colour. Give an exact answer. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Find

E(G)

, correct to the nearest cent. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

E(G)

= 0

. Find the new prize in dollars. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) With the original prizes, explain whether the game favours the player or the organisers over many games. [2]

0 words · a sentence or two

Late ferry crossingsShort response · 9 marks

Each crossing of a ferry to Kangaroo Island is late with probability

\frac{1}{5}

, independently of other crossings.

(a) Let

Y

= 1

if a particular crossing is late and

Y

= 0

otherwise. Find the standard deviation of

Y

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Let

X

be the number of late crossings among the next 4 crossings. Find

P(X

= 1)

. Give an exact answer. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

P(X

\ge 1)

. Give an exact answer. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Find the smallest number of crossings,

n

, for which the probability that none of them is late is less than 0.5. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Strong winds in Backstairs Passage often last for several days. Explain why the number of late crossings in a week might not be well modelled by a binomial distribution. [2]

0 words · a sentence or two

Germinating wildflower seedsShort response · 11 marks

A packet holds 20 seeds of a native wildflower. Each seed germinates with probability 0.85, independently of the other seeds. Let

X

be the number of seeds in a packet that germinate.

(a) Explain why

X

can be modelled by a binomial distribution. [2]

0 words · a sentence or two

(b) Find

P(X

= 17)

, correct to three significant figures. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Find the standard deviation of

X

, correct to three significant figures. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) A gardener buys 5 packets. Find the probability that at least 4 of the 5 packets each have at least 18 seeds germinate, correct to three significant figures. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

Shooting practiceShort response · 9 marks

A netball goal shooter scores each shot with probability 0.7, independently of her other shots.

(a) Let

Y

= 1

if she scores with her next shot and

Y

= 0

otherwise. Find the standard deviation of

Y

, correct to three significant figures. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) In a drill she takes 10 shots. Let

X

be the number she scores. Find

E(X)

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

P(X

= 7)

, correct to three significant figures. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Find

P(X

\ge 8)

, correct to three significant figures. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Given that she scores at least 7 goals in the drill, find the probability that she scores at least 8, correct to three significant figures. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

Cars per householdShort response · 9 marks

A survey of 250 households chosen at random in an Adelaide suburb recorded the number of cars,

X

, at each household. The results are shown in the table. The relative frequencies are used as estimates of the probability distribution of

X

Survey of 250 households, number of cars: 0 cars 20 households, 1 car 85, 2 cars 100, 3 cars 35, 4 cars 10.
Number of cars	0	1	2	3	4
Number of households	20	85	100	35	10

(a) Estimate

P(X

\ge 2)

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Estimate

E(X)

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

X

, correct to three significant figures. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Estimate the probability that

X

is within one standard deviation of its mean. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Explain why the probabilities found from this survey are only estimates of the probability distribution of

X

for the whole suburb. [2]

0 words · a sentence or two