DP IB Maths: AA HL

Topic Questions

4.4 Probability Distributions

1a
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2 marks

Frank plays a game involving a biased six-sided die.

The faces of the die are numbered 1 to 6.

The score of the game,X , is the number which lands face up after the die is rolled.

The following table shows the probability distribution for X.

Score,  bold italic x 1 2 3 4 5 6
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 1 over 6 1 half p 1 over 8 3 over 2 p 1 over 12 3 p

Calculate the exact value of p.

1b
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3 marks

Frank plays the game once.

Calculate the expected score.

1c
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3 marks

Frank plays the game twice and adds the scores together.

Find the probability Frank has a total score of 4, giving your answer as a fraction.

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2a
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1 mark

A weekly raffle ticket costs $ k, with three different levels of prizes, $ S.  The grand prize in the first week is $100 and it increases by $5 every week if nobody wins it.

The following table shows the probability distribution for S.

Prize,  bold italic s 0 20 Grand prize
bold italic P bold left parenthesis bold italic S bold equals bold italic s bold right parenthesis 9 p 7 p 4 p

Find the value of p.

2b
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2 marks

Given the grand prize is not won, write down an expression for the grand prize,G , in the

form G equals a plus b n, where a and b are constants to be found and n is the week of the raffle.

2c
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2 marks

Given the raffle is a fair game in the fourth week, find the value of k.

2d
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3 marks

Find the week in which the expected profit for the ticket buyer is $5.

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3a
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3 marks

A shooting target is divided in to three regions A, B and C.  Contestants pay $7.50 to enter and get to take one shot.

The probability of hitting each region is given in the following table:

Region A B C Missed target
Probability 1 over 15 2 over 15 a over 15 b over 15

It is given 3 a equals b and a comma b element of straight integer numbers.

Find the value of a and b.

3b
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4 marks

A contestant’s prize is dependent on the target they hit.

Region A B C Missed target
Prize $35 $k $ 7.50 0

Calculate the value of k spacesuch that the game is a fair game.

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4a
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3 marks

A game is played where contestants shoot a football at a goal with a goal keeper.  The goal is divided into five regions; straight A comma space straight B comma space straight C comma space straight D and straight E.  Each region is assigned with the scores, X, outlined in the table below.

ib4a-ai-sl-4-4-ib-maths-hard

The following table shows the value of  for each region and the probability distribution for X.

Region straight A straight B straight C straight D straight E Miss
Score  bold left parenthesis bold italic X bold right parenthesis 1 4 4 8 8 negative 2
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.3 p p q q 0.4

It is given that p equals 2 q

Find the exact value of p and q.

.

4b
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2 marks

Calculate the expected score.

4c
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3 marks

Find the probability that a player has a score of 16 after two rounds.

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5a
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2 marks

A spinner has six sections; A, B, C, D, E and F.  The table below shows the area of the spinner occupied by each section and their respective pay offs.

Section A B C D E F
Area 2 over 7 5 over 21 4 over 21 1 over 7 p 1 over 21
Prize $6 $5 $4 $3 $2 $1

Calculate the value of p.

5b
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6 marks

The game costs $4.

Calculate the expected profit from playing the game.

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6
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5 marks

A biased coin has a probability of showing tails as 0.85.  Leon plays a game where he flips the coin.  He pays $15  to play.  If the coin lands on tails he receives nothing but if it lands on heads he receives $ 5 c.  The game is fair.

Determine the value of c and write down the total prize if he wins.

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7a
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1 mark

Two biased coins are tossed and a fair spinner divided into four equal sectors numbered 1 to 4 is spun.

Write down the total number of possible outcomes when the two coins are tossed and the spinner is spun.

7b
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7 marks

A random variable, X, is defined as the number of tails when the two coins are tossed multiplied by the number the spinner lands on when it is spun.

For each coin the probability of getting tails is  1 over 6.

Complete the following probability distribution table for X:

bold italic x 0 1 2 3 4 6 8
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis              

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8a
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4 marks

A discrete random variable X has the following probability distribution:

bold italic x negative 5 negative 3 negative 1 0 1 3 5
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.24 2 k squared 0.04 0.12 3 k 0.19 0.11

Find the value of k.

8b
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3 marks

Find E left parenthesis X right parenthesis.

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9a
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2 marks

A discrete random variable X has the probability distribution shown in the following table:

 bold italic x 0 1 2 3 4 5 6
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis a 2 a 3 a 4 a 5 a 6 a 7 a

Find the value of a.

9b
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6 marks

Find:

(i)

P left parenthesis X less or equal than 4 right parenthesis

(ii)

P left parenthesis X greater or equal than 2 right parenthesis

(iii)

P left parenthesis 1 less than X less or equal than 5 right parenthesis

(iv)
P left parenthesis 0 less than X less than 6 right parenthesis

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10a
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2 marks

Tom has constructed a biased spinner with six sectors labelled 1 to 6.

The probability of the spinner landing on each of the six sectors is shown in the following table:

Number on sector 1 2 3 4 5 6
probability 1.5 p p 3 over 25 1 over 10 3 over 50 1 over 100

Find the value of p.

10b
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3 marks

Tom is playing a game with his biased spinner.  The score for the game, X, is the number the spinner lands on after being spun once.

Tom plays the game twice and adds the two scores together. Find the probability that the sum of Tom’s two scores is 9.

10c
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2 marks

Find the probability that X is:

(i)

no more than 3,

(ii)
at least 5.

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11a
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2 marks

5 cards numbered 1, 2, 5, 5, and 6 are identical when placed face down.  Throughout this question, a card is drawn randomly, the number is recorded and then the card is replaced and the five cards shuffled before a new card is taken.

Let X denote the value shown when one card is drawn.

Complete the following probability distribution table for X.

bold italic x        
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis        
11b
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2 marks

Find E open parentheses X close parentheses.

11c
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2 marks

Let Y represent the sum of the values when two cards are drawn.

Complete the following probability distribution table for Y.

bold italic y                  
bold italic P bold left parenthesis bold italic Y bold equals bold italic y bold right parenthesis                  
11d
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4 marks

Find E left square bracket Y right square bracket.

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1a
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1 mark

Three biased coins are tossed.

Write down all the possible outcomes when the three coins are tossed.

1b
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3 marks

A random variable,X , is defined as the number of heads when the three coins are tossed.

For each coin the probability of getting heads is  2 over 3 .

Complete the following probability distribution table for X:

bold italic x 0 1 2 3
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis        
1c
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2 marks

Represent the probability distribution for X as a piecewise function in the form 

P left parenthesis X equals x right parenthesis equals f left parenthesis x right parenthesis equals open curly brackets table row blank row blank end table close

1d
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2 marks

Represent the probability distribution for X as a bar chart.

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2
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2 marks

The random variable X has the probability function

P left parenthesis X equals x right parenthesis equals open curly brackets table row cell fraction numerator x over denominator 3 k end fraction end cell cell space space space space x equals 1 comma 2 comma 3 comma 4 comma 5 end cell row 0 cell space space space space otherwise end cell end table close

Show that  k equals 5.

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3a
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2 marks

The random variable X has the probability function

P left parenthesis X equals x right parenthesis equals open curly brackets table row cell k x end cell cell x equals 1 comma 3 comma 5 comma 7 end cell row cell 0 space 1 over k end cell otherwise end table close

Find the value of k.

3b
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2 marks

Find P left parenthesis X greater than 3 right parenthesis.

3c
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1 mark

State, with a reason, whether or not X is a discrete random variable.

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4a
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2 marks

The random variable X has the probability function

P open parentheses X equals x close parentheses equals open curly brackets table row cell 0.23 space space space space space space space space space x equals negative 1 comma 4 end cell row cell k space space space space space space space space space space space space space space space x equals 0 comma 2 end cell row cell 0.13 space space space space space space space space space x equals 1 comma 3 end cell row cell 0 space space space space space space space space space space space space space space space otherwise end cell end table close

Find the value of k.

4b
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2 marks

Construct a table giving the probability distribution of  X.

4c
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1 mark

Find P left parenthesis 0 less or equal than X less than 3 right parenthesis.

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5
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6 marks

A discrete random variable X has the probability distribution shown in the following table:

bold italic x 0 1 2 3 4
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 5 over 24 1 third 1 fourth 1 over 12 1 over 8

Find:

(i)

P left parenthesis X less than 4 right parenthesis

(ii)

P left parenthesis X greater than 1 right parenthesis

(iii)

P left parenthesis 2 less than X less or equal than 4 right parenthesis

(iv)
P left parenthesis 0 less than X less than 4 right parenthesis

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6a
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1 mark

Leonardo has constructed a biased spinner with six sectors labelled 0,1 , 2, 3, 4 and 5.  The probability of the spinner landing on each of the six sectors is shown in the following table:

number on sector 0 1 1 2 3 5
probability 6 over 20 p 3 over 20 5 over 20 3 over 20 1 over 20

Find the value of p.

6b
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3 marks

Leonardo is playing a game with his biased spinner.  The score for the game,X , is the number which the spinner lands on after being spun.

Leonardo plays the game twice and adds the two scores together. Find the probability that Leonardo has a total score of 5.

6c
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2 marks

Complete the following cumulative probability function table for X:

Score bold space bold italic x 0 1 2 3 5
bold italic P bold left parenthesis bold italic X bold less or equal than bold italic x bold right parenthesis 6 over 20       1

6d
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2 marks

Find the probability that is X

(i)

no more than 1

(ii)
at least 3.

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7a
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2 marks

A discrete random variable X has the following probability distribution:

bold italic x negative 3 negative 1 0 1 3
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.11 k squared 0.1 2 k 0.1

where k is a positive constant.

Show that k squared plus 2 k minus 0.69 equals 0.

7b
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1 mark

Hence find the value of k

7c
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3 marks

Find E left parenthesis X right parenthesis.

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8
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4 marks

A spinner is spun on a circle that is divided up into five sections, A comma B comma C comma D and E

The probability of the spinner landing on each section is given by the following table:

bold Region A B C D E
bold Probability 0.55 0.15 0.15 0.1 0.05

A person who rotates the spinner scores points depending on which section the spinner lands on.  These points are shown below.

bold Region A B C D E
bold Points negative 5 2 3 10 k

Given that the game is fair, find the value of k

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9a
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3 marks

A discrete random variable X has the following probability distribution:

bold italic x 0 1 2 3 4
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.1 0.05 a b 0.1

The value of  E open parentheses X close parenthesesequals 2.3.

Show that a spaceand b must satisfy the following two simultaneous equations:

a plus b equals 0.75

2 a plus 3 b equals 1.85

9b
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2 marks

Hence find the value of a spaceand the value of b .

9c
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2 marks

Find P left parenthesis 1 less or equal than X less than 4 right parenthesis.

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1a
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2 marks

Ben plays a game involving a biased eight-sided die.

The faces of the die are labelled negative 4 comma negative 2 comma negative 1 comma 0 comma 1 comma 3 comma 5 comma 6.

The score of the game, X, is the number which lands face up after the die is rolled.

The following table shows the probability distribution for X.

Score,  bold italic x negative 4 negative 2 negative 1 0 1 2 3 4
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 1 over 6 p 1 over 8 1 fourth 1 over 12 1 over 8 1 over 48 q

It is given that space p equals 4 q.

Calculate the exact values of p and q.

1b
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3 marks

Ben plays the game once.

Calculate the expected score.

1c
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3 marks

Ben plays the game twice and adds the scores together.

Find the probability that Ben has a total score of negative 3, giving your answer as a fraction in its simplest form.

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2a
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1 mark

A weekly lottery ticket costs $15, with five different levels of prizes,$ s .  The grand prize in the first week is $2000 and it increases by 20% each week that nobody wins it.

The following table shows the probability distribution for S.

Prize,  bold $ bold italic s 0 2 10 20 100 Grand prize
bold italic P bold left parenthesis bold italic S bold equals bold italic s bold right parenthesis 1 half 1 fourth 1 over 6 1 over 24 p 1 over 1000

Find the value of p.

2b
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2 marks

Determine if the lottery is a fair game in the first week.

2c
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2 marks

Given the grand prize is not won, write an expression in terms of n for the value of the grand prize in the nth week of the lottery.

2d
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4 marks

The wth week is the first week which a player is expected to make a profit.

Calculate the value of w spaceand the expected profit. Give your answer correct to 2 decimal places.

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3a
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4 marks

A discrete random variable X has the following probability distribution:

bold italic x negative 5 negative 3 negative 1 0 1 3 5
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.42 k to the power of 4 over 2 0.05 0.21 open parentheses fraction numerator 3 k over denominator 4 end fraction close parentheses squared 0.09 0.07

Find the value of k

3b
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3 marks

Find E open parentheses X close parentheses

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4a
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4 marks

A discrete random variable X has the following probability distribution:

bold italic x 0 1 2 3 4
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.04 0.35 a 0.21 b

The value of E left parenthesis X right parenthesis equals  2.28.

Write down two equations connecting a and b.

4b
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2 marks

Hence find the value of a and the value of b.

4c
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2 marks

Find P left parenthesis 1 less than X less or equal than 4 right parenthesis.

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5a
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5 marks

Two fair six-sided dice are rolled.  One is a standard die with sides numbered 1 to 6 and the other die has sides numbered 1, 1, 2, 2, 4, 4.

The discrete random variable S is the sum of these two dice when they are rolled.

Complete the following probability distribution table.

bold italic s                  
bold italic P bold left parenthesis bold italic S bold equals bold italic s bold right parenthesis                  
5b
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3 marks

Find E left parenthesis S right parenthesis.

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6a
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3 marks

Let X be the discrete random variable represented in the probability distribution table below.

bold italic x 0 1 2 3 4 5 6 7 8
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.32 0.22 0.21 1 over k 7 over k squared 4 over k squared 2 over k squared 1 over k squared 1 over k squared

Find the value of k.

6b
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3 marks

Find the expected value of X.

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7a
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1 mark

The table below represents the number of pets and the corresponding probability of a house having that number of pets.

Number of pets,  bold italic x 0 1 2 3 4
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.44 0.21 0.19 p 0.02

Find the value of p.

7b
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2 marks

Find the expected number of pets in a house.

7c
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4 marks

There was a recording error and houses with 5 pets were not counted.  It is found that there are 6 houses with 5 pets.  The neighbourhood in total has 406 houses, including these 6 houses.

Complete the following table for the true probability distribution of the number of pets, x.

Number of pets,  bold italic x 0 1 2 3 4 5
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis            
7d
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4 marks

Find the actual expected number of pets. Give your answer as a fraction in its simplest form.

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8a
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2 marks

The table below represents the probability distribution for the number of apple products people have in a city in France.

Number of apple products,  bold italic x bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis
0 0.1925
1 0.1815
2 0.2250
3 p
4 0.0895
5 0.0504
6 0.0307
7 0.0104
8 q

It is given that p equals 21 q

Find the value of p spaceand q.

8b
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2 marks

Find the expected number of apple products a randomly selected person from this city has.

8c
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2 marks

The city has a population of 412 000.  The average amount someone from this city spends on an apple product is €825.

Estimate the revenue apple earned from sales to people in this city. Give your answer to the nearest euro (€).

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9a
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2 marks

A discrete random variable X has the probability distribution shown in the following table:

bold italic x negative 5 negative 1 2 6
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 2 over 5 1 fourth p 4 p

Find the value of p.

9b
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6 marks

X is sampled twice such that the results of the two experiments are independent of each other and the outcomes of the two experiments are recorded.  A new random variable, Y, is defined as the sum of the two outcomes .

Draw a probability distribution table for Y.

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10a
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4 marks

The table below represents the number of strokes Josh takes on a particular round of golf and the corresponding frequencies over a year of playing at the same golf course.

Number of strokes, x Frequency
70 1
71 7
72 10
75 3
76 2
78 1
80 1

Complete the following probability distribution table for the data above.

Number of strokes, x P left parenthesis X equals x right parenthesis
70  
71  
72  
75  
76  
78  
80  

10b
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3 marks

Par refers to the number of strokes a golfer is expected to need to complete the play on a golf course.  The par number of strokes for Josh’s golf course is 72.

Determine whether Josh’s expected number of strokes is less than or greater than the par number of strokes.

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11a
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2 marks

Six cards numbered 4, 8, 8, 10, 10, and 15 are identical when placed face down.  Throughout this question a card is picked at random, the number on the front is recorded and then the card is replaced and the six cards are shuffled before a new card is taken.

Let X denote the value shown when one card is drawn.

Complete the following probability distribution table for X.

bold italic x        
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis        
11b
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2 marks

Find E left square bracket X right square bracket

11c
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2 marks

Let Y represent the product of the values when two cards are drawn.

Complete the following probability distribution table for Y.

bold italic y                    
bold italic P bold left parenthesis bold italic Y bold equals bold italic y bold right parenthesis                    
11d
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4 marks

Find E left square bracket Y right square bracket.

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