CIE AS Maths: Probability & Statistics 1

Topic Questions

3.4 Working with Distributions

1a
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4 marks

The table below shows five scenarios involving different random variables. Complete the table by placing a cross (×) in the correct box to indicate whether the random variable can be modelled by a binomial distribution, a normal distribution or neither. The first scenario is completed for you.

Scenario Binomial Normal Neither
The digits 1 to 9 are written on individual counters and placed in a bag.  A child randomly selects one of the nine counters at a time and replaces it after reading the number. The random variable A represents how many times the child selects a counter until the '5' is selected.     bold cross times
A farmer has many hens. The random variable B represents the mass of a randomly selected hen.      
A fair coin is flipped 100 times. The random variable C represents the number of times it lands on tails.      
A teacher has a 30-minute break for lunch. The random variable D represents the number of emails he receives during his lunch break.      
In a class of 30 students, each student rolls a fair six-sided dice with sides labelled 1 to 6. The random variable   represents the number of students who roll a number less than 5.      
1b
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1 mark

Write down the name of the probability distribution of A, the random variable described in part (a).

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2a
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2 marks

In an experiment there are a fixed number of trials and each trial results in a success or failure. Let X be the number of successful trials.  Write down the two other conditions that would need to be present to make X follow a binomial distribution.

2b
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3 marks

A fair spinner has 8 sectors labelled with the numbers 1 through 8. For each of the following cases, give a reason to explain why a binomial distribution would not be appropriate for modelling the specified random variable.

(i)
The random variable A is the number of times the spinner is spun until it lands on ‘1’ for the first time.

(ii)
When the spinner is spun it rotates exactly 115°. The random variable  is the number of times the spinner lands on ‘1’ when the spinner is spun 20 times.

(iii)
The random variable C is the sector number that the spinner lands on when it is spun once.
2c
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1 mark

State which one of the random variables defined in part (b) follows a geometric distribution.

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3a
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3 marks

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

(i)
100 red squirrels from the wild are sampled. The random variable A is the tail length of a randomly selected red squirrel.
(ii)
100 students sit a test which is marked out of 50. The random variable B is the number of marks achieved by a randomly selected student.
(iii)
100 men are in a shoe shop. The random variable C is the shoe size of a randomly selected man.
3b
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1 mark

The following histogram shows the distribution of results when a large number of measurements of the specified random variable D are made.  State with a reason whether a normal distribution would be appropriate for modelling the random variable.q3b-easy-4-4-choosing-distributions-edexcel-a-level-maths-statistics

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4a
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1 mark

The random variable X space tilde space B left parenthesis n comma p right parenthesis space can be approximated by Y space tilde space N left parenthesis mu comma sigma squared right parenthesis  when certain conditions are fulfilled.

State the condition for n which is required to use this approximation.

4b
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2 marks
(i)
State the value of p that will give the most accurate estimate.
(ii)
Give a reason to support your value.

 

4c
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5 marks

For each of the following binomial random variables, X:

  • state, with reasons, whether X can be approximated by a normal distribution
  • if appropriate, write down the normal approximation to X in the form  N left parenthesis mu comma sigma squared right parenthesis comma giving the values of mu and sigma squared.
(i)
X tilde B left parenthesis 6 comma 0.45 right parenthesis

(ii)
X tilde B left parenthesis 60 comma 0.05 right parenthesis

(iii)
X tilde B left parenthesis 60 comma 0.45 right parenthesis

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5a
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3 marks

The random variable X space tilde space B left parenthesis 100 comma 0.36 right parenthesis space is approximated by Y space tilde space N left parenthesis mu comma sigma squared right parenthesis.

Find the value of mu and show that sigma equals 4.8.

5b
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1 mark

Explain why a continuity correction must be incorporated when using this approximation.

5c
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4 marks

Use continuity corrections to find the value of k in each of the following approximations:

(i)
straight P left parenthesis X less or equal than 30 right parenthesis almost equal to straight P left parenthesis Y less than k right parenthesis

(ii)
straight P left parenthesis X less than 30 right parenthesis almost equal to straight P left parenthesis Y less than k right parenthesis

(iii)
straight P left parenthesis X greater or equal than 30 right parenthesis almost equal to straight P left parenthesis Y greater than k right parenthesis

(iv)
straight P left parenthesis X greater than 30 right parenthesis almost equal to straight P left parenthesis Y greater than k right parenthesis

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6a
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4 marks

X tilde B left parenthesis 75 comma 0.75 right parenthesis is approximated by a normal distribution Y space tilde space N left parenthesis mu comma sigma squared right parenthesis.

(i)
By calculating the values of n p and n open parentheses 1 minus p close parentheses comma show that n is sufficiently large enough so that a normal approximation is suitable.
(ii)
Find the values of mu and sigma.
6b
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7 marks
(i)
Calculate straight P open parentheses 55 less or equal than X less or equal than 57 close parentheses.
(ii)
Explain why straight P open parentheses 55 less or equal than X less or equal than 57 close parentheses space almost equal to space straight P open parentheses 54.5 less than straight Y less than 57.5 close parentheses.
(iii)
Calculate straight P open parentheses 54.5 space less than Y less than 57.5 close parentheses.

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1a
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4 marks

State the conditions that must be satisfied to be able to model a random variable X with a binomial distribution straight B left parenthesis n comma p right parenthesis.

1b
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4 marks

A fair spinner has 5 sectors labelled with the numbers 1 through 5. The spinner is spun and a fair coin is flipped, and the number the spinner lands on along with the result of the coin flip (heads or tails) are recorded.  For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

(i)
If the coin lands on heads, then the random variable S is the number of the sector that the spinner lands on times two.  Otherwise S is the number of the sector that the spinner lands on plus 10.
(ii)
The random variable Wis the number of times the spinner is spun and the coin is flipped until an odd number on the spinner occurs together with tails on the coin.
(iii)
The random variable Y is the number of times a prime number on the spinner occurs together with heads on the coin, when the spinner is spun and the coin is flipped 21 times.
(iv)
Each time the spinner is spun and the coin is flipped, it is a ‘win’ if a square number on the spinner occurs together with heads on the coin, or it is a ‘loss’ if a non-square number on the spinner occurs together with tails on the coin. Any other outcome is a ‘draw’.  The random variable L is the number of losses when the spinner is spun and the coin is flipped twelve times.
1c
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2 marks

For the random variable W defined in (b)(i) above, give the name of the probability distribution that would be appropriate for modelling W. Give the values(s) of any parameters.

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2a
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3 marks

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

(i)
A cake recipe calls for a certain amount of flour to be used. The random variable A is the number of cakes that can be made, following the recipe exactly each time, from a bag containing a random amount of flour.
(ii)
A student cuts a one-metre length of rope into two pieces at a random point. The random variable B is the difference in length between the two pieces of rope that result.
(iii)
People are chosen at random from the UK population. The random variable C is the age of a randomly selected person, measured from their date and time of birth.
2b
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4 marks

Each of the following histograms shows the distribution of results when a large number of measurements of the random variables DE or F are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable.  Where a normal model is appropriate, suggest a real-world variable that might show such a distribution.z-OoHfRM_q2b-medium-4-4-choosing-distributions-edexcel-a-level-maths-statistics

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3a
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1 mark

State the conditions that must be met for the distribution of a binomial random variable to be able to be approximated by a normal random variable.

3b
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5 marks

For each of the following binomial random variables, X tilde straight B left parenthesis n comma p right parenthesis:

  • if X can be approximated by a normal distribution, then write down the normal approximation to X in the form  straight N left parenthesis mu comma sigma squared right parenthesis, giving the values of mu and sigma,

  • if X cannot be approximated by a normal distribution, then give a reason why.
(i)
X tilde straight B left parenthesis 15 comma 0.5 right parenthesis

(ii)
X tilde straight B left parenthesis 150 comma 0.005 right parenthesis

(iii)
X tilde straight B left parenthesis 1500 comma 0.005 right parenthesis

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4a
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1 mark

The random variable  W tilde straight B left parenthesis 1200 comma 0.6 right parenthesis.

Give two reasons why a normal distribution can be used to approximate W.

4b
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7 marks

Find, using the appropriate normal approximation:

(i)
straight P left parenthesis 700 less than W less or equal than 730 right parenthesis

(ii)
straight P left parenthesis W greater or equal than 719 right parenthesis
4c
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4 marks

Using the normal approximation, find the largest value of k (where k is an integer) such that  straight P left parenthesis left parenthesis 720 minus k right parenthesis less than W less than left parenthesis 720 plus k right parenthesis right parenthesis less than 0.5.

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5a
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1 mark

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution B left parenthesis n comma p right parenthesis.

5b
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2 marks

On a European-style casino roulette wheel, the probability of the ball landing on a red number is 18 over 37.

The wheel is spun 36 times, and the ball lands on a red number X times.

Find straight P left parenthesis 17 less than X less or equal than 18 right parenthesis.

5c
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3 marks

In a separate experiment, the wheel is spun 1000 times and Y, the number of times the ball lands on a red number, is recorded.

(i)
Explain why a normal approximation would be appropriate in this case.
(ii)
Write down the normal distribution that could be used to approximate the distribution of Y.

 

5d
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4 marks

Use the distribution from (c)(ii) to approximate the probability that in 1000 spins the ball lands on a red number either less than 482 times or more than 491 times.

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6a
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2 marks

Due to a manufacturing irregularity, 41% of Adventure Dude action figures were produced with two left hands.  Although not especially rare, and therefore not especially collectible, these so-called ‘double left’ figures are nonetheless considered to be collector’s items by hard-core Adventure Dude fanatics.

A vintage toy shop has obtained 100 Adventure Dude action figures.  These may be assumed to represent a random sample.

Find the exact value of the probability that exactly 45 of the 100 figures are ‘double left’ figures. Give your answer to 6 decimal places.

6b
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4 marks

Use an appropriate normal approximation to approximate the probability that exactly 45 of the 100 figures are ‘double left’ figures.

6c
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2 marks

Find the percentage error when using your normal approximation from part (b) to estimate the probability that exactly 45 of the 100 figures are ‘double left’ figures. Give your answer correct to two decimal places.

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7a
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3 marks

The weight, W kg, of the feed in a sack of partridge feed produced by a certain manufacturer is modelled as  W tilde straight N left parenthesis 20 comma 0.01 right parenthesis.

Find straight P open parentheses W less than 19.75 close parentheses

7b
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3 marks

Roger buys ten sacks of the manufacturer’s partridge feed to feed to the partridges who have begun showing up at his backyard bird feeding station. 

Find the probability that all ten sacks contain feed with a weight that is within 250 space straight g space o f space 20 space kg.

7c
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4 marks

Find the probability that at least two of the ten sacks are not within 250 space straight g space o f space 20 space kg.

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1a
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4 marks

State the conditions that must be satisfied to be able to model a random variable X with a binomial distribution straight B left parenthesis n comma p right parenthesis.

1b
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4 marks

A fair spinner has 8 sectors labelled with the numbers 1 through 8. For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

(i)
The random variable S is the number of the sector that the spinner lands on when it is spun.
(ii)
The random variable W is the number of times the spinner is spun until it lands on ‘7’ for the first time.
(iii)
The random variable Y is the number of times the spinner lands on a prime number when it is spun twelve times.
(iv)
On the first spin, it is a ‘win’ if the spinner lands on an even number. On subsequent spins it is a ‘win’ if the spinner lands either on the same number as the previous spin or on a factor of the number from the previous spin.  The random variable   L is the number of wins when the spinner is spun ten times.
1c
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1 mark

For the random variable W defined in (b)(ii) above, give the name of the probability distribution that would be appropriate for modelling W.

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2a
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3 marks

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

(i)
A student cuts a one-metre length of rope into two pieces at a random point. The random variable A is the length of the shorter of these two pieces.
(ii)
You ask a sample of students in your school about their preferences for after-school activities. The random variable B is the number of students who say they prefer participating in lawn bowling.
(iii)
People are chosen at random from the UK population. The random variable C is the age of a randomly selected person, defined as the age which they turned on their most recent birthday.
2b
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3 marks

Each of the following histograms shows the distribution of results when a large number of measurements of the specified random variables –DE or  F– are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable.z-OoHfRM_q2b-medium-4-4-choosing-distributions-edexcel-a-level-maths-statistics

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3
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8 marks

For each of the following binomial random variables, X:

  • state, with reasons, whether X can be approximated by a normal distribution
  • if appropriate, write down the normal approximation to X in the form  N left parenthesis mu comma sigma squared right parenthesis, giving the values of mu and sigma.
(i)
X tilde straight B left parenthesis 8 comma 0.5 right parenthesis

(ii)
X tilde straight B left parenthesis 80 comma 0.54 right parenthesis

(iii)
X tilde straight B left parenthesis 625 comma 0.45 right parenthesis

(iv)
X tilde straight B left parenthesis 90 comma 0.95 right parenthesis

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4a
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1 mark

The random variable  W tilde B left parenthesis 500 comma 0.4 right parenthesis.

Explain why a normal distribution can be used to approximate W.

4b
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7 marks

Find, using the normal approximation:

(i)
straight P left parenthesis 189 less or equal than W less or equal than 211 right parenthesis
(ii)
straight P left parenthesis W greater than 220 right parenthesis
4c
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3 marks

Using the normal approximation, find the largest value of w such that text P end text left parenthesis W less or equal than w right parenthesis less than 0.1.

 

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5a
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1 mark

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution straight B left parenthesis n comma p right parenthesis.

5b
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2 marks

On a casino roulette wheel, the probability of the ball landing on a black number is  9 over 19 .

The wheel is spun 30 times, and the ball lands on a black number X times.

Find straight P left parenthesis X equals 14 right parenthesis.

5c
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3 marks

In a separate experiment, the wheel is spun 1000 times and Y, the number of times the ball lands on a black number, is recorded.

(i)
Explain why a normal approximation would be appropriate in this case.
(ii)
Write down the normal distribution that could be used to approximate the distribution of Y.
5d
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4 marks

Use the distribution from (c)(ii) to approximate the probability that in at least one half of the 1000 spins the ball lands on a black number.

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6a
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2 marks

As part of a marketing promotion, 47% of packets of a particular brand of crisps contain a zombie toy as a prize.  A random sample of 100 packets is taken.

Find the exact value of the probability that exactly 49 of the packets contain a prize. Give your answer to 6 decimal places. 

6b
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1 mark

Write down the normal distribution that could be used to approximate the distribution for the number of the 100 packets that contain a prize.

6c
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5 marks
(i)
Use the normal approximation to approximate the probability that exactly 49 of the packets contain a prize.
(ii)
Find the percentage error when using a normal approximation to calculate the probability that exactly 49 of the packets contain a prize. Give your answer correct to two significant figures.

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7a
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3 marks

The weight, W space straight g comma of a chocolate bar produced by a certain manufacturer is modelled as W tilde straight N open parentheses 200 comma 17.5 squared close parentheses.

Find straight P open parentheses W less than 195 close parentheses

7b
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2 marks

Hilda takes a random sample of chocolate bars and weighs them. Find the probability that the 8th chocolate bar that Hilda takes is the first one that weighs less than 195 space straight g.

7c
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2 marks

Zelda buys a pack containing 12 of the chocolate bars. It may be assumed that the 12 bars in the pack represent a random sample. Find the probability that exactly two chocolate bars in the pack weigh less than 195 space straight g.

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1a
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4 marks

State the conditions that must be satisfied to be able to model a random variable X with a binomial distribution space B left parenthesis n comma p right parenthesis.

1b
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4 marks

A fair spinner has 5 sectors labelled with the numbers 2, 3, 5, 7 and 11. A fair dice has 6 sides labelled with the numbers 1 through 6.  The spinner is spun and the dice is rolled, and the numbers that the spinner and the dice land on are recorded.  For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

(i)
The random variable S is the square root of the number that the spinner lands on, times the number that the dice lands on.
(ii)
The random variable W is the number of times that both the spinner and the dice land on a prime number, when the spinner is spun and the dice is rolled 47 times. 
(iii)
The random variable Y is the number of times the spinner is spun and the dice is rolled until the number the spinner lands on is a factor of the number the dice lands on. 
(iv)
On the first spin of the spinner and roll of the dice, it is a ‘win’ if the number on the spinner is greater than the number on the dice. On subsequent spins of the spinner and rolls of the dice, it is only a ‘win’ if the number the spinner lands on is higher than both the number the dice lands on and all the numbers that the dice has landed on previously.  The random variable Z  is the number of ‘wins’ when the spinner is spun and the dice is rolled 24 times.
1c
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2 marks

For the random variable Y defined in (b)(i) above, give the name of the probability distribution that would be appropriate for modelling Y. Give the values(s) of any parameters.

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2a
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3 marks

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

(i)
A string collector decides to measure all the pieces of string in his collection. The random variable A is the length of a randomly chosen piece of string from the collection, rounded to the nearest centimetre.
(ii)
A random sample of 1000 people is chosen from the US population, and the number of siblings each has is recorded. The random variable B is the mean number of siblings for the 1000 people in the sample. 
(iii)
The masses of individual eggs in the nests of a particular species of bird are measured and recorded. The random variable C is the total mass of the eggs in a randomly chosen nest.
2b
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6 marks

Each of the following histograms shows the distribution of results when a large number of measurements of the specified random variables –D, E or  F– are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable, and suggest a real-world variable that might show such a distribution.z-OoHfRM_q2b-medium-4-4-choosing-distributions-edexcel-a-level-maths-statistics

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3a
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1 mark

State the conditions that must be met for the distribution of a binomial random variable to be able to be approximated by a normal random variable.

3b
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5 marks

For each of the following binomial random variables, space X tilde B left parenthesis n comma p right parenthesis:

  • if X can be approximated by a normal distribution, then write down the normal approximation to X in the form  N left parenthesis mu comma sigma squared right parenthesis, giving the values of mu and sigma.
  • if X cannot be approximated by a normal distribution, then give a reason why
(i)
X tilde straight B left parenthesis 11 comma 0.5 right parenthesis

(ii)
X tilde straight B left parenthesis 40 comma 0.9 right parenthesis

(iii)
X tilde straight B left parenthesis 1300 comma 0.08 right parenthesis

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4a
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1 mark

The random variable  W tilde straight B left parenthesis 980 comma 0.4 right parenthesis.

 Explain why a normal distribution can be used to approximate W.

4b
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7 marks

Find, using the appropriate normal approximation:

(i)
straight P left parenthesis 386 less than W less than 398 right parenthesis
(ii)
straight P left parenthesis W greater or equal than 400 right parenthesis
4c
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4 marks

Using the appropriate normal approximation, find the smallest value of k(where k element of straight integer numbers)  such that  straight P left parenthesis k less than W less than left parenthesis 784 minus k right parenthesis right parenthesis less than 0.5.

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5a
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2 marks

On the roulette wheel at the Dunes Oddstacker casino, the probability of the ball landing on a red number is 6 over 13, and the probability of the ball landing on a black number is the same.  In general, the majority of bettors will lose their bets if the ball lands neither on a red number nor on a black number.

The wheel is spun 50 times, and the ball lands on a number that is either red or black X times.

Find straight P left parenthesis 44 less than X less than 46 right parenthesis.

5b
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1 mark

In a separate experiment, the wheel is spun 1000 times and Y, the number of times the ball lands neither on a red number nor on a black number, is recorded.

Explain why a normal approximation would be appropriate in this case.

5c
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5 marks

Use an appropriate normal distribution to approximate the probability that in 1000 spins the number of times the ball lands neither on a red number nor on a black number is neither 80 or more nor less than 75.

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6
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8 marks

Due to a production error, 58% of Bobbie Sue dolls were manufactured with proportions that might be seen on a real human being.  Plastic surgeons have been buying up these so-called ‘realie’ dolls, out of a concern that if too many of them are seen by the general public then numbers of people seeking plastic surgery will decrease.

A toy shop has received an order of 100 Bobbie Sue dolls.  These may be assumed to represent a random sample.

Calculate the percentage error when an appropriate normal approximation is used to calculate probability that exactly 58 of thr 100 dolls are 'realie' dolls.

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7
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9 marks

A machine is used to produce the barge poles sold by the You Would Touch It With One Of Ours barge pole company.  The actual length, L m, of the barge poles is normally distributed with mean mu m, standard deviation sigma m, and interquartile range 0.01712 m. 

Twenty of the barge poles are chosen at random. 

Find the probability that at most two of the barge poles will be shorter than the mean by 1 cm or more.

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