Distributions - IB Math AA Question Bank

Question 1

SL • Paper 1

Easy

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SL • Paper 1

Easy

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A discrete random variable $X$ has probability distribution

P(X=x)=k(x+1), \quad x\in\{0,1,2,3\}

A

Find the value of $k$ .

[2]

B

Find $E(X)$ .

[2]

C

A player receives $2X$ counters and pays an entry fee of $c$ counters to play. Find $c$ if the game is fair.

[2]

Question 2

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

The random variable $X$ is normally distributed with mean $50$ . It is known that approximately $68\%$ of the values of $X$ lie between $44$ and $56$ .

A normal curve centred at its mean, with a symmetric central region shaded between two vertical boundary lines. The centre and the two boundary values are labelled as given in the question, and the shaded area is labelled as approximately 68 percent.

A

Find the standard deviation of $X$ .

[2]

B

Find the approximate value of $P(X>56)$ .

[2]

C

Using the empirical rule, write down the approximate value of $P(38<X<62)$ .

[1]

Question 3

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

A discrete random variable $X$ has probability distribution

P(X=x)=k(x+1), \quad x\in\{0,1,2,3,4\}

In a game, a player receives a prize of $1.50 for each point scored by $X$ .

A

Find the value of $k$ .

[2]

B

Find $E(X)$ .

[2]

C

Find the entry fee that would make the game fair.

[1]

Question 4

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

A discrete random variable $X$ has probability distribution

P(X=x)=k(6-x), \quad x\in\{1,2,3,4,5\}

A

Find $k$ .

[2]

B

Find $P(X\geq 3)$ .

[1]

C

A score of $X$ gives a prize of $ $2X$ . Find the fair entry fee for this game.

[2]

Question 5

SL • Paper 1

Medium

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SL • Paper 1

Medium

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Let $X\sim B\left(4,\frac{1}{3}\right)$ .

A

Find $P(X=2)$ .

[2]

B

Find $P(X\geq 1)$ .

[2]

C

Find $\operatorname{Var}(X)$ .

[1]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A box contains a large number of identical components. The probability that a randomly selected component is defective is $\frac{1}{5}$ . A quality inspector selects $5$ components independently. Let $X$ be the number of defective components selected.

A

State two assumptions needed for $X$ to be modelled by a binomial distribution.

[2]

B

Find $P(X=1)$ .

[2]

C

Find the expected number of defective components selected.

[1]

Question 7

SL • Paper 1

Medium

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SL • Paper 1

Medium

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The random variable $X$ is normally distributed with mean $30$ . It is given that $P(X<24)=0.18$ .

A

Write down $P(X>36)$ .

[1]

B

Find $P(24<X<36)$ .

[2]

C

Find $P(X<36\mid X>24)$ .

[2]

Question 8

HL • Paper 1

Medium

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HL • Paper 1

Medium

Non Calculator

The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ . The value $68$ has standardized value $-1$ , and the value $92$ has standardized value $2$ .

A

Find $\mu$ and $\sigma$ .

[3]

B

Find the value of $x$ whose standardized value is $\frac{1}{2}$ .

[1]

C

Let $Z\sim N(0,1)$ and suppose $P\left(Z<\frac{1}{2}\right)=q$ . Write down $P(X>80)$ in terms of $q$ .

[1]

Question 9

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The discrete random variable $X$ has probability distribution

\begin{array}{c|ccc} x & -1 & 0 & 2\\ \hline P(X=x) & \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \end{array}

A

Find $E(X)$ .

[2]

B

Find $\operatorname{Var}(X)$ .

[3]

C

Let $Y=4-2X$ . Find $\operatorname{Var}(Y)$ .

[1]

Question 10

HL • Paper 1

Medium

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HL • Paper 1

Medium

Non Calculator

The continuous random variable $X$ has probability density function

f(x)= \begin{cases} kx, & 0\leq x\leq 2,\\ 0, & \text{otherwise.} \end{cases}

A

Find the value of $k$ .

[2]

B

Find $P(X>1)$ .

[2]

C

Find $E(X)$ .

[2]

Question 11

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A packet contains $12$ seeds. Each seed germinates independently with probability $0.82$ . Let $X$ be the number of seeds in the packet that germinate.

A

Write down the distribution of $X$ .

[1]

B

Find $P(X=10)$ .

[2]

C

Find the probability that at least $9$ seeds germinate.

[2]

D

Find the standard deviation of $X$ .

[1]

Question 12

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The lifetime, $T$ hours, of a type of rechargeable battery is modelled by $T\sim N(\mu,7.5^2)$ . It is known that $18\%$ of these batteries last less than $42$ hours.

A

Determine the value of $\mu$ .

[3]

B

Using your value of $\mu$ , find $P(45<T<55)$ .

[2]

Question 13

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A discrete random variable $X$ has probability distribution

P(X=x)=cx^2, \quad x\in\{1,2,3,4\}

A new random variable is defined by $Y=5-2X$ .

A

Find $c$ .

[1]

B

Find $E(X)$ .

[2]

C

Find $\operatorname{Var}(X)$ .

[2]

D

Find $E(Y)$ and $\operatorname{Var}(Y)$ .

[1]

Question 14

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The continuous random variable $X$ has probability density function

f(x)=\begin{cases} ax, & 0\leq x<2,\\ a(4-x), & 2\leq x\leq 4,\\ 0, & \text{otherwise}. \end{cases}

A

Find $a$ .

[2]

B

Find $E(X)$ .

[2]

C

Find $\operatorname{Var}(X)$ .

[2]

Question 15

SL • Paper 1

Medium

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SL • Paper 1

Medium

Non Calculator

A discrete random variable $X$ can take the values $1$ , $2$ and $4$ . Its probability distribution is

P(X=1)=p,\quad P(X=2)=2p,\quad P(X=4)=1-3p

It is given that $E(X)=3$ .

A

Find $p$ .

[3]

B

Find $P(X\geq 2)$ .

[1]

C

In a game, a player receives $X$ counters. Find the entry fee, in counters, that makes the game fair.

[1]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The continuous random variable $X$ has probability density function

f(x)= \begin{cases} ax, & 0\leq x\leq 1,\\ a(2-x), & 1<x\leq 2,\\ 0, & \text{otherwise.} \end{cases}

A

Find $a$ .

[2]

B

Write down the mode of $X$ .

[1]

C

Show that the median of $X$ is $1$ .

[2]

Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The continuous random variable $X$ has probability density function

f(x)= \begin{cases} 3x^2, & 0\leq x\leq 1,\\ 0, & \text{otherwise.} \end{cases}

A

Find $E(X)$ .

[2]

B

Find $\operatorname{Var}(X)$ .

[2]

C

Let $Y=6X-1$ . Find $E(Y)$ and $\operatorname{Var}(Y)$ .

[2]

Question 18

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $X\sim N(40,16)$ and $Y\sim N(50,25)$ . Let $Z\sim N(0,1)$ , and suppose $P\left(Z<\frac{3}{2}\right)=q$ .

A

Find the standardized value corresponding to $X=46$ .

[1]

B

Express $P(X>46)$ in terms of $q$ .

[1]

C

Find the value of $y$ such that $P(Y<y)=P(X>46)$ .

[3]

Question 19

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The mass, $M$ grams, of apples from a farm is modelled by a normal distribution with mean $165$ and standard deviation $12$ .

A

Find $P(150<M<180)$ .

[2]

B

Find the $90$ th percentile of the apple masses.

[2]

C

A box contains $20$ independently selected apples from the farm. Find the probability that at least $18$ of them have mass greater than $150$ grams.

[2]

Question 20

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A factory produces electronic components. The probability that a component is defective is $0.04$ , independently of other components. A box contains $n$ components.

A

Write down an expression, in terms of $n$ , for the probability that a box contains at least one defective component.

[2]

B

Find the smallest value of $n$ such that the probability that a box contains at least one defective component is greater than $0.5$ .

[2]

C

For this value of $n$ , find the expected number of defective components in a box.

[1]

Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ . It is known that

P(X<72)=0.12 \quad \text{and} \quad P(X>95)=0.08

A

Write down two equations involving $\mu$ and $\sigma$ .

[3]

B

Determine $\mu$ and $\sigma$ .

[2]

C

Find the standardized value corresponding to $X=90$ .

[1]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The continuous random variable $X$ has probability density function

f(x)=\begin{cases} kx(4-x), & 0\leq x\leq 4,\\ 0, & \text{otherwise}. \end{cases}

A

Find $k$ .

[2]

B

Find $P(X>3)$ .

[2]

C

Find the lower quartile of $X$ .

[2]

Question 23

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The continuous random variable $X$ has probability density function

f(x)=\begin{cases} ke^{-0.2x}, & x\geq 0,\\ 0, & x<0. \end{cases}

A

Show that $k=0.2$ .

[2]

B

Find $P(X>8)$ .

[2]

C

Find the value of $m$ such that $P(X<m)=0.90$ .

[2]

Question 24

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The marks on a standardized test are modelled by $X\sim N(68,9^2)$ . Students with a standardized value greater than $1.2$ receive an award. A school has $50$ students taking the test, and their marks may be assumed independent.

A

Assuming that marks are whole numbers, find the minimum mark required to receive an award.

[2]

B

Find the probability that a randomly selected student receives an award.

[1]

C

Find the probability that at least $8$ of the $50$ students receive an award.

[2]

D

Find the expected number of students in the school who receive an award.

[1]

Question 25

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A discrete random variable $X$ has probability distribution

\begin{array}{c|cccc} x&0&1&2&4\\ \hline P(X=x)&p&2p&3p&1-6p \end{array}

where $p$ is a constant.

A

Find the range of possible values of $p$ .

[2]

B

I.

Given that $E(X)=2$ , find the value of $p$ .

[2]

II.

Hence find $P(X\geq 2)$ .

[1]

III.

Two independent observations of $X$ are made. Find the probability that their sum is at least $4$ . If you did not obtain a value of $p$ in part (b)(i), use $p=\frac{1}{8}$ .

[2]

C

In a game, a player receives $3X$ counters. Find the entry fee, in counters, that would make the game fair.

[3]

Question 26

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A player takes five independent shots at a target. The probability of hitting the target on each shot is $p$ . Let $X$ be the number of hits. It is given that

P(X=0)=32P(X=5)

A

State the distribution of $X$ .

[1]

B

I.

Find $p$ .

[3]

II.

Hence find $P(X=2)$ .

[2]

C

Find the expected number of hits and the variance of $X$ . If you did not obtain a value of $p$ , use $p=\frac{1}{3}$ .

[2]

D

The player receives $12$ points if at least one shot hits the target, and receives $0$ points otherwise. Find the fair entry fee, in points, for this game.

[2]

Question 27

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The random variable $X$ is normally distributed with mean $80$ . It is known that approximately $95\%$ of the values of $X$ lie between $68$ and $92$ .

A

Find the standard deviation of $X$ .

[2]

B

I.

Using the empirical rule, write down the approximate value of $P(74<X<86)$ .

[1]

II.

Using the empirical rule, find the approximate value of $P(X>92)$ .

[2]

III.

Using the empirical rule, find the approximate value of $P(68<X<86)$ .

[1]

C

Two independent values of $X$ are selected. Find the approximate probability that both values are greater than $92$ .

[3]

Question 28

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The length, $L$ cm, of a manufactured rod is modelled by a normal distribution with mean $20$ . Using the empirical rule, approximately $95\%$ of rods have lengths between $18$ cm and $22$ cm. A rod is called short if its length is less than $19$ cm.

A

Find the standard deviation of $L$ .

[2]

B

I.

Using the empirical rule, find the approximate probability that a rod is short.

[2]

II.

Find the approximate probability that a rod has length greater than $22$ cm.

[1]

C

Three rods are selected independently. Find the approximate probability that exactly one of them is short. If you did not obtain an answer in part (b)(i), use $P(\text{short})=0.16$ .

[2]

D

Give two reasons why the probability found in part (c) may not be a reliable prediction for a real production line.

[2]

Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

At a school fair a game gives a score $X$ . The probability distribution of $X$ is shown in the table.

\begin{array}{c|cccc} x&0&2&5&8\\ \hline P(X=x)&0.18&p&0.22&q \end{array}

It is known that $E(X)=3.70$ .

A

I.

Write down an equation involving $p$ and $q$ .

[1]

II.

Find the values of $p$ and $q$ .

[3]

B

I.

Find $P(X\ge 5)$ .

[1]

II.

The game is played twice independently. Find the probability that the total score is at least $10$ . If you did not obtain values for $p$ and $q$ , use $p=\frac{11}{30}$ and $q=\frac{7}{30}$ .

[3]

C

In a different game using the same score $X$ , a player receives $3 if $X=5$ , receives $10 if $X=8$ , and receives nothing otherwise. Find the entry fee that would make this game fair.

[3]

Question 30

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

An airline finds that each passenger who buys a ticket for a particular short flight independently has probability $0.08$ of not arriving for the flight.

A

For a flight with $18$ tickets sold, let $X$ be the number of passengers who do not arrive.

I.

State the distribution of $X$ .

[1]

II.

Find $P(X=2)$ .

[2]

III.

Find the expected number of passengers who do not arrive.

[2]

B

For a separate flight scenario, the aircraft has $18$ seats. The airline decides to sell $20$ tickets. Let $Y$ be the number of passengers who arrive for the flight.

I.

State the distribution of $Y$ .

[1]

II.

Find the probability that more passengers arrive than there are seats.

[2]

III.

The airline pays $60 compensation for each passenger who arrives but cannot be seated. Find the expected compensation paid on one such flight.

[2]

C

State one reason why the binomial model for the number of passengers who arrive may not be appropriate in this context.

[2]

Question 31

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The time $T$ minutes taken by a courier to complete a delivery route is modelled by $T\sim N(32,4.5^2)$ . A route taking more than $38$ minutes is called late.

A

I.

Find the probability that a route is late.

[2]

II.

Twenty-five routes are completed independently. Find the probability that at most one of them is late. If you did not obtain an answer to part (a)(i), use $0.0912$ .

[3]

B

I.

Find the $95$ th percentile of the delivery route times.

[2]

II.

The company pays a $5 refund for a late route and an additional $10 refund if the route takes more than $42$ minutes. Find the expected refund for $100$ independent routes.

[2]

C

Give one reason why the normal model may not be suitable for delivery times on all days.

[2]

Question 32

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A new website advertises a subscription service. Each visitor independently has probability $0.035$ of buying a subscription. In a campaign with $120$ independent visitors, let $X$ be the number of visitors who buy a subscription.

A

In one advertising campaign there are $120$ visitors. Let $X$ be the number of visitors who buy a subscription.

I.

State the distribution of $X$ .

[1]

II.

Find $P(X\le 3)$ .

[2]

III.

Find $E(X)$ and $\operatorname{Var}(X)$ .

[2]

B

I.

Each subscription gives the website revenue of $18. Find the expected revenue from the $120$ visitors.

[1]

II.

The campaign has a fixed cost of $c. Find $c$ if the expected profit is zero.

[2]

C

Find the smallest number of independent visitors required so that the probability of at least one subscription is greater than $0.95$ .

[2]

Question 33

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Two spinners, $A$ and $B$ , are spun independently. Let $A$ be the score on the first spinner, with probability distribution

\begin{array}{c|cccc} a&1&2&3&4\\ \hline P(A=a)&\frac{1}{2}&\frac{1}{4}&\frac{1}{8}&\frac{1}{8} \end{array}

Let $B$ be the score on the second spinner, where

P(B=1)=q,\quad P(B=2)=2q,\quad P(B=5)=1-3q

A

Find $E(A)$ .

[2]

B

I.

Find the range of possible values of $q$ .

[2]

II.

Find $E(B)$ in terms of $q$ .

[2]

C

It is given that $P(B>A)=\frac12$ . Find $q$ .

[3]

D

A game is played by spinning both spinners once. The player's gain is $B-A-c$ , where $c$ is an entry fee. Find $c$ so that the game is fair. If you did not obtain $q$ , use $q=\frac14$ .

[3]

Question 34

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

A student answers $n$ independent multiple-choice questions. The probability that the student answers any one question correctly is $p$ . Let $X$ be the number of correct answers. It is given that

E(X)=3\quad\text{and}\quad \operatorname{Var}(X)=2

A

I.

Write down two equations involving $n$ and $p$ .

[2]

II.

Hence find $n$ and $p$ .

[2]

B

Find the probability that the student answers no questions correctly. If you did not obtain values for $n$ and $p$ , use $n=9$ and $p=\frac13$ .

[2]

C

A prize of $5$ points is awarded for each correct answer, but a fixed penalty of $d$ points is deducted. Find $d$ so that the expected score is $8$ points.

[2]

D

State two assumptions, other than independence, needed for this binomial model to be appropriate.

[2]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ . The value $35$ has standardized value $-2$ , and the value $50$ has standardized value $1$ . Let $Z\sim N(0,1)$ and let $P(Z<1)=q$ .

A

I.

Write down two equations involving $\mu$ and $\sigma$ .

[2]

II.

Hence find $\mu$ and $\sigma$ .

[2]

B

Express $P(40<X<50)$ in terms of $q$ .

[3]

C

Find the value of $x$ such that $P(X>x)=P(X<35)$ .

[2]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A discrete random variable $X$ has probability distribution

\begin{array}{c|cccc} x&-1&1&2&4\\ \hline P(X=x)&p&2p&p&1-4p \end{array}

It is given that $E(X)=2$ .

A

I.

Find $p$ .

[2]

II.

Find $E(X^2)$ . If you did not obtain $p$ , use $p=\frac{2}{13}$ .

[3]

B

Find $\operatorname{Var}(X)$ .

[2]

C

Let $Y=3-2X$ . Find $E(Y)$ and $\operatorname{Var}(Y)$ .

[2]

D

Find the possible values of $a$ such that $\operatorname{Var}(aX+5)=42$ .

[2]

Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A continuous random variable $X$ has probability density function

f(x)=\begin{cases} a\sin x,&0\leq x\leq \pi,\\ 0,&\text{otherwise.} \end{cases}

A

I.

Show that $a=\frac12$ .

[2]

II.

State the mode of $X$ .

[1]

III.

Find $P\left(X<\frac{\pi}{6}\right)$ .

[1]

B

Show that the median of $X$ is $\frac{\pi}{2}$ .

[2]

C

Find $E(X)$ .

[2]

D

Give a reason why the answers to parts (b) and (c) are equal.

[2]

Question 38

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A cafe uses two machines to fill cups of coffee. The volume $A$ ml from machine A is normally distributed with mean $250$ and standard deviation $6$ . The volume $B$ ml from machine B is normally distributed with mean $255$ and standard deviation $5$ . On a particular day, $70\%$ of cups are filled by machine A and the rest by machine B. A cup is selected at random.

A

I.

Find the probability that a cup filled by machine A contains less than $245$ ml.

[2]

II.

Find the probability that the selected cup contains less than $245$ ml.

[2]

B

I.

Given that the selected cup contains less than $245$ ml, find the probability that it was filled by machine A.

[3]

II.

Twelve cups are selected independently from all cups filled that day. Find the probability that at least three contain less than $245$ ml. If you did not obtain an answer to part (a)(ii), use $0.148$ .

[2]

C

The cafe wants to label the largest $10\%$ of cups filled by machine A as extra-full. Find the minimum volume for this label.

[3]

Question 39

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The length $L$ cm of a fish in a lake is modelled by $L\sim N(34,3.2^2)$ . A fish is classified as undersized if $L<30$ , and large if $L>38$ .

A

(a)

I.

Find the probability that a randomly selected fish is undersized.

[2]

II.

Find the probability that a fish has length between $30$ cm and $38$ cm.

[2]

B

(b)

I.

In a sample of $40$ independently selected fish, find the probability that fewer than three are undersized. If you did not obtain an answer to part (a)(i), use $0.106$ .

[3]

II.

Find the expected number of large fish in $200$ independently selected fish.

[1]

C

Find the value of $k$ such that $P(34-k<L<34+k)=0.90$ .

[3]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The time $T$ minutes taken by runners in a long race is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$ . It is known that $P(T<210)=0.18$ and $P(T<260)=0.92$ .

A

I.

Write down two equations involving $\mu$ and $\sigma$ using standardized values.

[2]

II.

Hence find $\mu$ and $\sigma$ .

[3]

B

I.

Find $P(220<T<250)$ . If you did not obtain values for $\mu$ and $\sigma$ , use $\mu=229.72$ and $\sigma=21.55$ for subsequent calculations; otherwise use your unrounded values.

[2]

II.

A runner is in the fastest $5\%$ if their time is less than $r$ minutes. Find $r$ to 3 significant figures. If you did not obtain values for $\mu$ and $\sigma$ , use $\mu=229.72$ and $\sigma=21.55$ for this calculation; otherwise use your unrounded values.

[2]

C

Sixteen runners are selected independently. Find the probability that at least four of them finish in less than $210$ minutes.

[3]

Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The number of claims $N$ made by a customer in one year has probability distribution

\begin{array}{c|cccc} n&0&1&2&5\\ \hline P(N=n)&a&3a&0.25&0.15 \end{array}

The annual cost to an insurer, in dollars, is $C=200N+80$ .

A

I.

Find $a$ .

[2]

II.

Find $E(N)$ .

[1]

III.

Find $\operatorname{Var}(N)$ .

[2]

B

I.

Find $E(C)$ .

[2]

II.

Find $\operatorname{Var}(C)$ . If you did not obtain $\operatorname{Var}(N)$ , use $2.31$ .

[2]

C

A premium $P$ dollars is charged. A loss is made by the insurer if $C>P$ .

I.

Find the premium that makes the expected profit zero.

[1]

II.

Determine the smallest whole-number premium such that the probability of a loss is at most $0.20$ .

[2]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Readings $R$ from an environmental sensor are modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$ . It is known that

P(R<18)=0.10,\qquad P(R>30)=0.05

A

I.

Write down two equations involving $\mu$ and $\sigma$ using standardized values.

[2]

II.

Determine $\mu$ and $\sigma$ .

[3]

B

I.

Find the standardized value corresponding to $R=27$ .

[2]

II.

An alert is triggered when $R>27$ . Find the probability that an alert is triggered.

[2]

C

I.

Over $30$ independent days, find the probability that alerts are triggered on at least $8$ days. If you did not obtain an answer to part (b)(ii), use $0.180$ .

[2]

II.

Determine the alert threshold $h$ such that the expected number of alerts in $30$ independent days is $3$ .

[1]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A game uses a spinner which gives a score $X\in\{0,1,2,3\}$ . The probability of score $x$ is modelled by

P(X=x)=k(x+\theta), \quad x=0,1,2,3

where $\theta>0$ . A player pays an entry fee and then receives two tokens for each point scored.

Score, $x$	Probability, $P(X=x)$
0	$k\theta$
1	$k(1+\theta)$
2	$k(2+\theta)$
3	$k(3+\theta)$

A

I.

Show that $k=\dfrac{1}{6+4\theta}$ .

[2]

II.

Find $E(X)$ in terms of $\theta$ .

[2]

B

It is found from many trials that the mean score is approximately $1.90$ . Determine the value of $\theta$ predicted by the model.

[2]

C

I.

Find the entry fee, in tokens, which makes the original game fair.

[1]

II.

For this value of $\theta$ , find the variance of the player's net gain when the fair entry fee is used.

[3]

D

A modified game charges $3.50$ tokens but limits the number of tokens received to a maximum of $3$ . Determine whether this modified game is favourable to the player in the long run.

[3]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A security system consists of independent sensors. In one installation there are $8$ sensors, each of which detects movement with probability $p$ . The probability that at least one sensor detects movement is $0.760$ .

A schematic diagram of a room with several identical sensors positioned around it. Each sensor has the same detection probability and sensors are labelled as acting independently.

A

I.

Show that $p\approx 0.163$ .

[2]

II.

State the distribution of $Y$ , the number of detections in $20$ independent sensors of the same type.

[1]

III.

Find $E(Y)$ .

[1]

B

Find the probability that at least $4$ of the $20$ sensors detect movement.

[2]

C

Determine the smallest number of independent sensors required so that the probability of at least one detection is greater than $0.99$ .

[3]

D

Comment on one modelling assumption that may be invalid if the sensors are placed very close together.

[2]

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The volume $X$ ml in bottles filled by a machine is modelled by a normal distribution $N(\mu,\sigma^2)$ . It is known that $P(X<490)=0.04$ and $P(X>515)=0.10$ .

Normal density of bottle volumes with cut lines at 490 ml and 515 ml.

A

I.

Write down two equations involving $\mu$ and $\sigma$ .

[2]

II.

Determine $\mu$ and $\sigma$ .

[3]

B

Find $P(500<X<510)$ .

[2]

C

Twelve bottles are selected independently. Find the probability that at least $7$ have volumes between $500$ ml and $510$ ml. If you did not obtain an answer to part (b), use $0.455$ .

[3]

D

Find the value of $a$ such that $P(\mu-a<X<\mu+a)=0.98$ .

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A continuous random variable $X$ has probability density function

f(x)=\begin{cases} kx(6-x), & 0\le x\le 6,\\ 0, & \text{otherwise.} \end{cases}

Unscaled qualitative sketch of the parabolic shape of f(x) on its support.

A

I.

Show that $k=\dfrac{1}{36}$ .

[2]

II.

Find $P(X>4)$ .

[2]

B

Find $E(X)$ and $\operatorname{Var}(X)$ .

[4]

C

Explain why the median of $X$ is $3$ .

[2]

D

A new random variable is defined by $Y=10-2X$ . Find $E(Y)$ and $\operatorname{Var}(Y)$ .

[2]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A website records whether it has at least one service interruption on each day. The events on different days are modelled as independent. From historical data, the probability of no service interruptions in a period of $30$ days is $0.214$ .

Context	Length / days	Relevant data
Historical 30-day period	30	$P(\text{no interruption in all 30 days})=0.214$
Future 90-day period	90	$X=$ number of days with at least one interruption
Model assumption	—	Days are independent; daily interruption probability (p) is constant

A

I.

Show that the model gives a daily interruption probability of approximately $0.0501$ .

[2]

II.

For a future period of $90$ days, state the distribution of $X$ , the number of days with at least one interruption.

[1]

III.

Find the mean number of days with interruptions in this $90$ -day period.

[1]

B

Find the probability that there are fewer than $3$ days with interruptions in a future $90$ -day period. Give your answer to 3 significant figures.

[2]

C

Determine the smallest integer $k$ such that $P(X\le k)\ge 0.95$ .

[3]

D

The website manager suspects that after one interruption, another interruption is more likely on the following day. Explain how this would affect the suitability of the binomial model.

[2]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An insurance policy covers the number of claims $N$ made by one customer in a year. The distribution is shown in the table.

\begin{array}{c|ccccc} n&0&1&2&3&4\\ \hline P(N=n)&a&2a&3a&2a&a \end{array}

The annual cost to the insurer, in units, is $C=120N+50$ .

n	P(N=n)	C [units]
0	a	50
1	2a	170
2	3a	290
3	2a	410
4	a	530

A

I.

Find $a$ .

[1]

II.

Find $E(N)$ and $\operatorname{Var}(N)$ .

[4]

B

Find $E(C)$ and $\operatorname{Var}(C)$ .

[3]

C

A premium $P$ is charged. Find the premium which makes the expected annual profit zero.

[1]

D

The insurer wants the probability of making a loss on this customer to be at most $0.20$ . Determine the smallest premium, in whole units, which achieves this.

[2]

Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Scores in two aptitude tests are modelled by normal distributions. Test A scores are modelled by $A\sim N(82,6^2)$ . Test B scores are modelled by $B\sim N(\mu,\sigma^2)$ . For Test B, $P(B<74)=0.08$ and $P(B>94)=0.15$ .

Normal score distributions for Tests A and B.

A

I.

Find the standardized value corresponding to a Test A score of $90$ .

[1]

II.

Find $P(A>90)$ .

[2]

B

Determine $\mu$ and $\sigma$ for Test B.

[4]

C

A certificate is awarded to the top $10\%$ of Test B candidates. Find the minimum Test B score required for a certificate.

[2]

D

A student scores $92$ on Test A and $92$ on Test B. Determine on which test the student performed better relative to the test population. Justify your answer.

[3]

Question 50

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The length $L$ mm of a manufactured part is modelled by $L\sim N(100,2.5^2)$ . A part is classified as acceptable if $96<L<104$ . A batch contains independently produced parts.

Normal density of part lengths with limits at 96 and 104 mm.

A

I.

Find the probability that a randomly selected part is acceptable.

[2]

II.

State the distribution of $X$ , the number of acceptable parts in a batch of $15$ parts.

[1]

III.

Find $E(X)$ .

[1]

B

Find the probability that at least $14$ parts in a batch of $15$ are acceptable.

[2]

C

Determine the smallest batch size $n$ for which the expected number of unacceptable parts is greater than $5$ .

[2]

D

Find the greatest batch size $n$ for which the probability that all parts are acceptable is greater than $0.5$ .

[2]

Question 51

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A training app gives a user $20$ independent questions. For this user, the probability of answering any one question correctly is $0.72$ . Let $X$ be the number of correct answers.

A simple interface-style diagram showing a sequence of identical quiz questions, each with two possible outcomes, correct or incorrect. The diagram emphasizes independent trials with the same success probability.

A

I.

State the distribution of $X$ .

[1]

II.

Find $E(X)$ and $\operatorname{Var}(X)$ .

[2]

B

Find $P(X\ge 16)$ .

[2]

C

A badge is awarded if the user answers at least $r$ questions correctly. Determine the smallest value of $r$ such that the probability of receiving the badge is less than $0.10$ .

[3]

D

The app charges an entry fee of $e$ tokens and gives a prize of $50$ tokens if the badge is awarded. For $r=18$ , find the fair entry fee.

[2]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A continuous random variable $X$ has probability density function

f(x)=\begin{cases} k(1+x),&0\leq x\leq 2,\\ 0,&\text{otherwise.} \end{cases}

A

I.

Find $k$ .

[2]

II.

Find the cumulative distribution function $F(x)$ for $0\leq x\leq 2$ .

[3]

B

Find the median of $X$ .

[3]

C

Find $E(X)$ and $\operatorname{Var}(X)$ .

[3]

Question 53

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A continuous random variable $T$ has probability density function

f(t)=\begin{cases} ke^{-t},&t\geq 0,\\ 0,&t<0. \end{cases}

A

I.

Show that $k=1$ .

[2]

II.

Find $P(T>a)$ , where $a\geq 0$ .

[2]

III.

Find the median of $T$ .

[1]

B

Find $E(T)$ .

[2]

C

Given that $E(T^2)=2$ , find $\operatorname{Var}(T)$ .

[2]

D

A machine is replaced at time $c$ . Find $c$ such that the probability that the machine lasts longer than $c$ is equal to the median of $T$ divided by $4$ .

[3]

Question 54

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ . The value $72$ has standardized value $-1$ , and the value $84$ has standardized value $\frac12$ . Let $Z\sim N(0,1)$ , with $P(Z<1)=a$ and $P(Z<2)=b$ .

A

I.

Write down two equations involving $\mu$ and $\sigma$ .

[2]

II.

Hence find $\mu$ and $\sigma$ .

[2]

B

Express $P(64<X<88)$ in terms of $a$ and $b$ .

[3]

C

Four independent observations of $X$ are made. Find, in terms of $a$ , the probability that exactly three of the observations are greater than $88$ .

[3]

Question 55

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The continuous random variable $X$ has probability density function

f(x)=\begin{cases} c(x+1),&0\le x\le 2,\\ c(5-x),&2<x\le 5,\\ 0,&\text{otherwise.} \end{cases}

Probability density function of X on its support.

A

I.

Show that $c=\dfrac{2}{17}$ .

[3]

II.

State the mode of $X$ .

[1]

III.

Find $P(1<X<3)$ .

[1]

B

Find the median of $X$ .

[4]

C

I.

Find $E(X)$ .

[2]

II.

Find $\operatorname{Var}(X)$ . If you did not obtain an answer to part (c)(i), use $E(X)=2.14$ .

[3]

Question 56

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $a>-1$ , the continuous random variable $X$ has probability density function

f(x)=\begin{cases} k(1+ax^2),&0\le x\le 1,\\ 0,&\text{otherwise.} \end{cases}

It is given that $E(X)=0.580$ .

A

I.

Show that $k=\dfrac{1}{1+\frac{a}{3}}$ .

[2]

II.

Find the value of $a$ .

[3]

B

Use $a=\dfrac{24}{17}$ for this part.

I.

Find $P(X<0.4)$ .

[2]

II.

State the mode of $X$ , giving a reason.

[2]

C

I.

Find the median of $X$ .

[2]

II.

Find $\operatorname{Var}(X)$ . If you did not obtain a value for $a$ , use $a=1.41$ .

[2]

Question 57

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The continuous random variable $X$ has probability density function

f(x)=\begin{cases} k\left(4-(x-1)^2\right),&-1\le x\le 3,\\ 0,&\text{otherwise.} \end{cases}

A new random variable is defined by $Y=50+10X$ .

Probability density function of X on [-1, 3], a symmetric downward-opening parabola.

A

I.

Show that $k=\dfrac{3}{32}$ .

[3]

II.

Find $P(X>2)$ .

[2]

B

I.

Find $E(X)$ , giving a reason.

[2]

II.

Find $\operatorname{Var}(X)$ .

[2]

C

I.

Find $E(Y)$ and $\operatorname{Var}(Y)$ .

[2]

II.

Find the value of $c$ such that $P(Y>c)=0.100$ .

[2]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $t>-1$ , a continuous random variable $X$ has density

f_t(x)=\begin{cases} c(1+tx), & 0\le x\le 1,\\ 0, & \text{otherwise.} \end{cases}

This family is used to model a measurement that may be biased towards larger or smaller values.

Three linear densities from the same family.

A

I.

Find $c$ in terms of $t$ .

[2]

II.

Find $E(X)$ in terms of $t$ .

[2]

B

It is known that the median is $0.600$ . Determine $t$ .

[3]

C

For this value of $t$ , find the mode and the mean.

[2]

D

Prove that, for this family, $E(X)>\dfrac12$ if and only if $t>0$ .

[2]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The waiting time $X$ minutes for a response from an automated system has probability density function

f(x)=\begin{cases} k(x+1)e^{-x}, & x\ge 0,\\ 0, & x<0. \end{cases}

Probability density curve for waiting time X, shown without the exact intercept.

A

I.

Show that $k=\dfrac12$ .

[2]

II.

Find the cumulative distribution function $F(x)$ for $x\ge 0$ .

[2]

B

Find $E(X)$ and $\operatorname{Var}(X)$ .

[4]

C

Determine the median waiting time.

[2]

D

A response taking more than $3$ minutes is described as delayed. Find the probability that, in $10$ independent responses, at least one is delayed.

[2]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a>1$ , a continuous random variable $X$ has probability density function

f(x)=\begin{cases} kx, & 0\le x<1,\\ k, & 1\le x<a,\\ k(a+1-x), & a\le x\le a+1,\\ 0, & \text{otherwise.} \end{cases}

This density has a flat central section and two sloping sides.

x interval	f(x)
0 ≤ x < 1	kx
1 ≤ x < a	k
a ≤ x ≤ a+1	k(a+1-x)
otherwise	0

A

I.

Show that $k=\dfrac1a$ .

[2]

II.

Explain why the distribution is symmetric about $x=\dfrac{a+1}{2}$ .

[2]

B

It is known that $E(X)=2.5$ . Determine $a$ .

[2]

C

For $a=4$ , find $P(X<1.5)$ .

[2]

D

For $a=4$ , find $\operatorname{Var}(X)$ .

[3]

E

State the median when $a=4$ , giving a reason.

[2]