AQA AS Maths: Statistics

Topic Questions

4.1 Probability Distributions

1a
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3 marks

The discrete random variable, X, is defined as the number of sixes obtained from rolling two fair dice.

(i)
Find the probability of obtaining two sixes from rolling two fair dice.

(ii)
Complete the following probability distribution table for X:

bold italic x 0 1 2
bold italic P stretchy left parenthesis X equals x stretchy right parenthesis 25 over 36    
1b
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2 marks

Use the table, or otherwise, to find the probability of obtaining at least one six from rolling two fair dice.

 

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2a
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1 mark

 The discrete random variable X  has the probability function

straight P open parentheses X equals x close parentheses space equals open curly brackets table row cell 1 fourth space space space space space space x equals 0 comma 1 comma 2 comma 3 end cell row cell 0 space space space space space space space space otherwise end cell end table close space

Briefly explain why has a uniform probability distribution.

2b
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2 marks

Find:

(i)
straight P left parenthesis 1 less or equal than X less or equal than 2 right parenthesis

(ii)
straight P left parenthesis X less than 3 right parenthesis.

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3a
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2 marks

The discrete random variable has the probability function

straight P open parentheses X equals x close parentheses space equals open curly brackets table row cell k x space space space space space space space x equals 2 comma 3 end cell row cell 0 space space space space space space space space space otherwise end cell end table close

Use the fact that the sum of all probabilities equals 1 to show that k equals 0.2.

3b
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2 marks

Write down:

(i)
straight P left parenthesis 2 less or equal than X less than 3 right parenthesis

(ii)
straight P left parenthesis X equals 5 right parenthesis

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4a
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2 marks

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

bold italic x 2 4 6 8 10
2 over 5 1 over 10 1 fifth p 1 over 10

Use the fact that the sum of all probabilities equals 1 to find the value of p.

4b
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4 marks

Find:

(i)
straight P left parenthesis X less or equal than 4 right parenthesis

(ii)
straight P left parenthesis X greater than 7 right parenthesis

(iii)
straight P left parenthesis 2 less or equal than X less or equal than 6 right parenthesis

(iv)
text P end text left parenthesis 3 less than X less than 7 right parenthesis

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5a
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2 marks

The discrete random variable  has the probability function

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

 Use the fact that the sum of all probabilities equals 1 to show that k equals 1 over 7.

5b
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1 mark

Briefly explain why has a non-uniform probability distribution.

5c
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2 marks

Show that straight P left parenthesis X less or equal than 2 right parenthesis equals straight P left parenthesis X equals 4 right parenthesis.

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6a
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2 marks

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

bold italic x 1 2 3 4 5
Error converting from MathML to accessible text. 5 over 12 2 over 12 1 over 12 3 over 12 1 over 12


Complete the following cumulative probability function table for X :

bold italic x 1 2 3 4 5
Error converting from MathML to accessible text. 5 over 12 7 over 12     1
6b
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5 marks

Use your table from part (a) to find:

(i)
straight P left parenthesis X less or equal than 3 right parenthesis

(ii)
straight P left parenthesis X greater or equal than 4 right parenthesis

(iii)
straight P left parenthesis 2 less or equal than X less or equal than 4 right parenthesis

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7a
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3 marks

The discrete random variable has the cumulative probability distribution shown in the following table:

bold italic x -2 -1 0 1 2
bold P stretchy left parenthesis X equals x stretchy right parenthesis 1 fifth 2 over 5 3 over 5 4 over 5 5 over 5

Complete the following probability distribution table for X:

bold italic x -2 -1 0 1 2
Error converting from MathML to accessible text. 1 fifth 1 fifth      
7b
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2 marks

Find:

(i)
straight P left parenthesis X less than 0 right parenthesis

(ii)
straight P left parenthesis X greater than 0 right parenthesis.
7c
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2 marks

Explain, with a reason, whether has a uniform probability distribution or not.

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8a
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1 mark

The discrete random variable  has the probability function

straight P open parentheses X equals x close parentheses equals open curly brackets table row cell 1 fourth space space space space space space space space space space space space x equals 0 end cell row cell 1 over 8 space space space space space space space space space space space space x equals 1 comma 2 space end cell row cell 5 over 16 space space space space space space space space space x equals 3 end cell row cell p space space space space space space space space space space space space space x equals 4 end cell row cell 0 space space space space space space space space space space italic space italic space space otherwise end cell end table close

Briefly explain how you can deduce that  p equals 3 over 16.

8b
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2 marks

Find straight P left parenthesis 1 less or equal than X less or equal than 2 right parenthesis.

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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 minus the number of tails.

Given that for each coin the probability of getting heads is 3 over 5,

complete the following probability distribution table for X:

bold italic x        
bold italic P begin bold style stretchy left parenthesis X equals x stretchy right parenthesis end style        
1c
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2 marks

Represent the probability distribution for X as a probability mass function.

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2
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3 marks

The random variable  X has the probability function

P open parentheses X equals x close parentheses equals open curly brackets table row cell 1 over k space space space space space space space space space space space space x equals 1 comma 2 comma 3 comma 5 comma 8 comma 13 end cell row cell 0 space space space space space space space space space space space space space space otherwise end cell end table close

(i)
Find the value of k.

(ii)
Write down the name of this probability distribution.

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3a
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2 marks

A student claims that a random variable has a probability distribution defined by the following probability mass function:

P open parentheses X equals x close parentheses equals open curly brackets table row cell x squared over 30 space space space space space space space space space space space space space space space space space space x equals 1 comma 1 comma 3 comma 5 end cell row cell 0 space space space space space space space space space space space space space space space space space space space space space space otherwise end cell end table close space space space space space space space space space space space

Explain how you know that the student’s function does not describe a probability distribution.

3b
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2 marks

Given that the correct probability mass function is of the form

P open parentheses X equals x close parentheses equals open curly brackets table row cell x squared over k space space space space space space space space space space space space space space space x equals negative 1 comma 1 comma 3 comma 5 end cell row cell 0 space space space space space space space space space space space space space space space space space space space otherwise end cell end table close

where k is a constant,

find the exact value of k.

3c
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2 marks

Find straight P left parenthesis X greater than 0 right parenthesis.

3d
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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 attributes columnalign left end attributes row cell 0.21 space space space space space space space space space space space space space x equals 0 comma 1 end cell row cell k x space space space space space space space space space space space space space space space space x equals 3 comma 6 end cell row cell 0.11 space space space space space space space space space space space space x equals 10 comma 15 end cell row cell 0 space space 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 straight P left parenthesis 3 less than X less or equal than 14 right parenthesis

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5a
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1 mark

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

bold italic x -1 1 2
bold P begin bold style stretchy left parenthesis X equals x stretchy right parenthesis end style 5 over 12 p 1 fourth


Find the value of p.

5b
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5 marks

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.

Complete the following probability distribution table for Y:

bold italic y -2 0 1 2 3 4
bold P stretchy left parenthesis Y equals y stretchy right parenthesis            
5c
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4 marks

Find:

(i)
straight P left parenthesis Y not equal to 0 right parenthesis

(ii)
straight P left parenthesis Y greater than 1 right parenthesis

(iii)
straight P left parenthesis negative 2 less than Y less than 2 right parenthesis

(iv)
straight P left parenthesis Y less than 0 space space o r space space Y greater or equal than 2 right parenthesis

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6a
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4 marks

Leonidas is playing a game with a fair six-sided dice on which the faces are numbered 1 to 6.  He rolls the dice until either a ‘6’ appears or he has rolled the dice four times.  The random variable X is defined as the number of times that the dice is rolled.

 Write down the probability distribution of X in table form.

6b
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2 marks

Complete the following cumulative probability function table for X:

bold italic x 1 2 3 4
bold P bold left parenthesis bold italic X bold less or equal than bold italic x bold right parenthesis        
6c
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2 marks

Find the probability that X is

(i)
at most 3

(ii)
at least 3.

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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.

Given that for each coin the probability of getting heads is  2 over 3 ,

complete the following probability distribution table for X:

x 0 1 2 3
P(X=x)        
1c
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2 marks

represent the probability distribution for X as a probability mass function.

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2
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3 marks

The random variable X  has the probability function

straight P open parentheses X equals x close parentheses equals open curly brackets table row cell 1 over k space space space space space space space x equals 1 comma 2 comma 3 comma 4 comma 5 end cell row cell 0 space space space space space space space space space space otherwise end cell end table close

(i)
Show that  k = 5.

(ii)
Write down the name of this probability distribution.

 

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3a
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2 marks

The random variable X has the probability function

straight P open parentheses X equals x close parentheses equals space open curly brackets table row cell k x space space space space space space space space space space space space space space x equals 1 comma 3 comma 5 comma 7 end cell row cell 0 space space space space space space space space space space space space space space space space otherwise end cell end table close space

 Find the value of k.

3b
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2 marks

Find straight 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

straight P open parentheses X equals x close parentheses equals space 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 straight P open parentheses 0 less or equal than X less than 3 close parentheses.

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5
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4 marks

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

x 0 1 2 3 4
P(X = x) 5 over 24 1 third 1 fourth 1 over 12 1 over 8


Find:

(i)
straight P left parenthesis X less than 4 right parenthesis

(ii)
straight P left parenthesis X greater than 1 right parenthesis

(iii)
straight P left parenthesis 2 less than X less or equal than 4 right parenthesis

(iv)
straight 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, 1, 2, 3 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 italic x 0 1 2 3 5
bold 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 X is

(i)
no more than 1

(ii)
at least 3.

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1a
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1 mark

Two biased coins are tossed and a fair spinner with three sectors numbered 1 to 3 is spun.

Write down all the possible outcomes when the two coins are tossed and the spinner is spun.

1b
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5 marks

A random variable, , is defined as the number of heads 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 heads is  1 third.

Complete the following probability distribution table for X:

bold italic x 0 1 2 3 4 6
           
1c
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2 marks

Represent the probability distribution for X as a probability mass function.

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2
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3 marks

The random variable X can take the values k squared left parenthesis negative 1 right parenthesis to the power of k  for  k equals 0 comma space 2 comma space 3 comma space 5 comma space 6.

Given that  X  is distributed as a discrete uniform distribution, write down the probability mass function of X.

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3a
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2 marks

A student claims that a random variable X has a probability distribution defined by the following probability mass function:

P open parentheses X equals x close parentheses equals open curly brackets table row cell fraction numerator 1 over denominator 3 x squared end fraction space space space space space space space space space space space space space space space space x equals negative 3 comma negative 1 end cell row cell fraction numerator 1 over denominator 3 x cubed end fraction space space space space space space space space 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 space space space space space space space space otherwise end cell end table close

Explain how you know that the student’s function does not describe a probability distribution.

3b
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2 marks

Given that the correct probability mass function is of the form

 P open parentheses X equals x close parentheses equals open curly brackets table row cell k over x squared space space space space space space space space space space space space space space space space space x equals negative 3 comma negative 1 end cell row cell k over x cubed space space space space space space space space 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 space space space space space space otherwise end cell end table close

where k is a constant,

 Find the exact value of k.

3c
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2 marks

Find straight P left parenthesis X less than 2 right parenthesis.

3d
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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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4 marks

The random variable X has the probability function

P open parentheses X equals x close parentheses equals x squared over 495 comma space space space space space space space x equals p comma 2 p comma 3 p comma 4 p comma 5 p

where  p greater than 0 space spaceis a constant.

 Construct a table giving the probability distribution of X.

4b
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2 marks

Complete the following cumulative probability function table for X:

bold italic x          
bold P bold left parenthesis bold italic X bold less or equal than bold italic x bold right parenthesis         1
4c
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3 marks

Find:

(i)
straight P left parenthesis 3 less than X less or equal than 12 right parenthesis

(ii)
the probability that X is no more than 10

(iii)
the probability that X is at least 10.

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5
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6 marks

The independent random variables X  and have probability distributions

                             straight P left parenthesis X equals x right parenthesis equals p comma space space space space space space x equals 1 comma 2 comma 3 comma 5 comma 8 comma 11 space

                         straight P left parenthesis Y equals y right parenthesis equals q over y comma space space space space space space y equals 1 comma 3 comma 6 space

where p and q are constants.

 Find  straight P left parenthesis X greater than Y right parenthesis.

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6a
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6 marks

Leofranc is playing a gambling game with a fair six-sided dice on which the faces are numbered 1 to 6.  He must pay £2 to play the game.  He then chooses a ‘lucky number’ between 1 and 6, and rolls the dice until either his lucky number appears or he has rolled the dice four times.  If his lucky number appears on the first roll, he receives £5 back.  If his lucky number appears on the second, third or fourth rolls, he receives £3, £2 or £1 back respectively.  If his lucky number has not appeared by the fourth roll, then the game is over and he receives nothing back.

 The random variable W is defined to be Leofranc’s profit (i.e., the amount of money he receives back minus the cost of playing the game) when he plays the game one time.  Note that a negative profit indicates that Leofranc has lost money on the game.

Write down the probability distribution of W in table form.

6b
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3 marks

Find the probability that when playing the game one time Leofranc

(i)
wins money

(ii)
loses money

(iii)
breaks even (i.e., does not lose money).

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