Further Probability (Edexcel A Level Maths: Statistics): Exam Questions

The Venn diagram, where and are probabilities, shows the three events , and and their associated probabilities.

The events and are independent.

Find the value of and the value of

Find

A company has 1825 employees.

The employees are classified as professional, skilled or elementary.

The following table shows

• the number of employees in each classification

• the two areas, or , where the employees live

Some classifications of employees are more likely to work from home.

• 65% of professional employees in both area and area work from home

• 40% of skilled employees in both area and area work from home

• 5% of elementary employees in both area and area work from home

• Event F is that the employee is a professional

• Event H is that the employee works from home

• Event R is that the employee is from area A

Using this information, complete the Venn diagram below.

Find

Find

Find

Three bags, , and , each contain 1 red marble and some green marbles.

• Bag contains 1 red marble and 9 green marbles only

• Bag contains 1 red marble and 4 green marbles only

• Bag contains 1 red marble and 2 green marbles only

Sasha selects at random one marble from bag .

If he selects a red marble, he stops selecting.

If the marble is green, he continues by selecting at random one marble from bag .

If he selects a red marble, he stops selecting.

If the marble is green, he continues by selecting at random one marble from bag .

Draw a tree diagram to represent this information.

Given that Sasha selects a red marble, find the probability that he selects it from bag .

and are three events with  = 0.2, ,  and    .

Given that events and  are mutually exclusive, and that events and  are independent, draw a Venn diagram to illustrate the probabilities.

Find:

(i)

(ii)

(iii)

240 students are surveyed regarding their belief in supernatural creatures.  144 say they believe in unicorns  .  75 say they believe in vampires .  Of those who believe in vampires, 27 also believe in unicorns.

Draw a two-way table to show this information.

One student is chosen at random.  Find:

(i)

(ii)

(iii)

(iv)

and are two events with  and    Given that and  are independent, write down

(I)

(ii)

A group of middle and senior school students were asked whether they preferred vinegar or ketchup as a topping on their chips. The following two-way table shows the results of the survey:

(i) Find and P(ketchup|middle) and P(middle|ketchup).

(ii) Use your results from part (b)(i) to show that for the students in the sample ‘is in middle school’ and ‘prefers ketchup on chips’ are independent events.

The following Venn diagram shows the number of adults in a poll who said they enjoy watching action films (A), Bollywood musicals (B), and crime thrillers (C).

One of the adults who was polled is selected at random.  Given that the adult chosen enjoys watching at least one of those three genres of film, find the probability that the adult enjoys watching:

(i) Bollywood musicals

(ii) only one of the three genres of film

(iii) exactly two of the three genres of film.

Three events  and , are such that B and C are mutually exclusive and A and C are independent. ,    and  .

Given that , draw a Venn diagram to show the probabilities for events , and .

Find:

(i)

(ii)

(iii)

Given that  ,    and  , find:

(i)

(ii)

The event C has  . The events A and C are mutually exclusive. 

Given that , find .

A bag contains 15 blue tokens and 27 yellow tokens.  A token is taken from the bag and its colour is recorded, but it is not replaced in the bag.  A second token is then taken from the bag and its colour is recorded.

 Draw a tree diagram to represent this information.

Find the probability that:

(i) the second token selected is blue

(ii) both tokens selected are blue, given that the second token selected is blue. 

Ichabod is a keen chess player who plays one game of chess online every night before going to bed.  In any one of those games, the probabilities of Ichabod winning, drawing, or losing are 0.4, 0.27 and 0.33 respectively.  Following each game, the probabilities of Ichabod sleeping well after winning, drawing or losing are 0.7, 0.9 and 0.2 respectively.

 Draw a tree diagram to represent this information.

Find the probability that on a randomly chosen night

(i) Ichabod loses his chess game and sleeps well

(ii) Ichabod sleeps well.

Given that Ichabod sleeps well, find the probability that his chess game did not end in a draw.

The Venn diagram shows the probabilities associated with four events, , , and .

Given that , find the value of .

Given also that and are independent, find the value of .

Given further that , find

(i) the value of ,

(ii) the value of .

A large college produces three magazines.

One magazine is about green issues, one is about equality and one is about sports.

A student at the college is selected at random and the events , and are defined as follows

is the event that the student reads the magazine about green issues

is the event that the student reads the magazine about equality

is the event that the student reads the magazine about sports

The Venn diagram, where , , and are probabilities, gives the probability for each subset.

Find the proportion of students in the college who read exactly one of these magazines.

No students read all three magazines and

Find

(i) the value of

(ii) the value of

Given that , find

(i) the value of

(ii) the value of

Determine whether or not the events and are independent.

Show your working clearly.

A,B  and C are three events with  , ,  and   .

Given that events A and C are mutually exclusive, and that events B and C are independent, draw a Venn diagram to illustrate the probabilities.

Find:

(i)

(ii)

(iii)

Some attendees at a pizza fandom convention are surveyed regarding their opinions about anchovies and bananas as pizza toppings.  144 of them say they do not like anchovies.  320 of them say they do not like bananas.  28 of them say they like bananas but not anchovies.  Only 12 of them like both toppings.

Let A and B be the events ‘likes anchovies as a pizza topping’ and ‘likes bananas as a pizza topping’ respectively.

 Draw a two-way table to show this information.

One of the attendees is chosen at random. Find:

(i)

(ii)

(iii)

(iv)

and are two events with x  and  y,  where x0.  Given that and  are independent, find the following probabilities in terms of x and y:

(i)

(ii)

A group of 18- to 25-year-olds, and a group of people over 65 years old, were asked whether they would prefer to holiday in Ibiza or Skegness. The following two-way table shows part of the results of the survey:

(i) Find P(over 65) and P(Ibiza)

(ii) Given that for the people in the sample the events ‘is over 65’ and ‘prefers to holiday in Ibiza’ are independent, find the missing values and complete the table of survey results.  

The following Venn diagram shows some of the results for the number of chess players in a poll who said they enjoy playing any, all or none of three chess game openings – the Najdorf Sicilian (N), the Orangutan (O), and the Ponziani countergambit (P).

120 chess players were polled in total, of whom one half said they enjoy playing the Najdorf Sicilian, one third said they enjoy playing the Orangutan, and one fourth said they enjoy playing the Ponziani countergambit. 

 Use the above information to fill in the missing values in the Venn diagram.

One of the chess players who was polled is selected at random. 

Given that the chess player chosen enjoys playing at least one of the three openings, find the probability that the chess player enjoys playing:

(i) the Orangutan

(ii) only one of the three openings

(iii) at least two of the three openings

 Three events,,  and , are such that and  are independent and  and  are mutually exclusive. ,  , and the following two relations also hold: 

Using the above information, draw a Venn diagram to show the probabilities for events , and .

Find:

(i)

(ii)

(iii)

Given that  ,   and  , find:

(i)

(ii)

The event  has . The events  and  are mutually exclusive. 

Given that  and  are independent, find  .

A bag contains 12 orange marbles, 8 purple marbles and 5 red marbles.  A marble is taken from the bag and its colour is recorded, but it is not replaced in the bag.  A second marble is then taken from the bag and its colour is recorded.

Draw a tree diagram to represent this information.

Find the probability that:

(i) both marbles are different colours

(ii) the second marble is purple, given that both marbles are different colours.

Rosco is a somewhat inept rural county sheriff who frequently finds himself involved in car chases with well-meaning local entrepreneurs.  During any given car chase, Rosco inevitably runs into one of three obstacles – a damaged bridge (with probability 0.47), an oil slick (with probability 0.32), or a pigpen at the end of a dead-end road.

If he encounters a damaged bridge there is a 25% chance that he will make it across safely; otherwise he lands in the river and ends up covered in mud.  If he encounters an oil slick there is a 40% chance that his car will spin around and he will end up continuing his hot pursuit in the wrong direction; otherwise he goes off the road into a farm pond and ends up covered in mud.  If he encounters a pigpen at the end of a dead-end road there is a 15% chance he will stop his car in time; otherwise he drives into the muddy end of the pigpen while the pigs sit at the other end laughing.  If he drives into the muddy end of a pigpen there is a 20% chance he will only end up covered in mud; otherwise he ends up covered in mud and other things that are found in pigpens.

Draw a tree diagram to represent this information.

Find the probability that in the course of a randomly chosen car chase

(i) Rosco ends up covered in mud

(ii) Rosco ends up covered in mud, but only in mud.

Given that Rosco ends up covered in mud in the course of a randomly chosen car chase, find the probability that he didn’t encounter an oil slick. Give your answer as an exact value.

In the course of a particular day Rosco finds himself engaged in three separate car chases with well-meaning local entrepreneurs.  The car chases may be considered to be independent events.

Determine the probability that on that day Rosco will not end up covered in other things that are found in pigpens.

, and  are three events such that ,   and  In addition the following two relations hold:

Draw a Venn diagram to illustrate the probabilities.

Find:

(i)

(ii)

(iii)

The discrete random variable X  has probability function

x =1,2,3,4,5

Find the value of k.

Draw a two-way table of probabilities for the events ‘X  is odd’ (O) and ‘X  is prime’ (P).

Find:

(i)

(ii)

(iii)

and  are two events with  and  . 
Given that  and  are independent, show that

A group of maths students and maths teachers were asked whether they preferred reading Latin poetry or partaking in extreme sports during their free time. The following two-way table shows part of the results of the survey:

Given that for the teachers and students in the sample the events ‘is a maths student’ and ‘prefers to read Latin poetry during free time’ are independent, find the missing values and complete the table of survey results.

A lifestyle magazine for ancient historians recently conducted a poll of its readership.  The following Venn diagram shows some of the results for the numbers of the historians polled who said that they were fans of any, all or none of the following three ancient Near Eastern rulers – the Hittite king Suppiluliuma I (S),  the Assyrian king Ashurbanipal (A), and the Akkadian king Manishtushu (M).

180 historians were polled in total.  For each historian who was only a fan of Manishtushu, there were 7 who were only fans of Suppiluliuma I.  For each two historians who were fans of both Ashurbanipal and Manishtushu but not of Suppiluliuma I, there were nine historians who were fans of all three rulers.  Most of the historians were, as expected, fans of at least one of the three rulers, but it was also discovered that being a fan of Ashurbanipal and being a fan of Manishtushu were independent events.

One of the historians who was polled is selected at random.  Given that the historian chosen is a fan of at least one of the three rulers, find the exact probability that the historian is a fan of: 

(i) Suppiluliuma I

(ii) at least two the three rulers

(iii) exactly two of the three rulers.

Events and  are such that    and  .

 Given that , find the range of possible values of  .

Additionally, event  is such that   and  .

Determine whether this additional information would change your answer to part (a), and if it would then give the modified range of possible values of  when this additional information is taken into account.

Given that , find the range of possible values of .

and  are events such that  ,  and ,  where    and  .

Find the following probabilities in terms of and :

(i)

(ii)

(iii)

A bag contains tokens that each have a single number written on them.  The integers between 1 and 15 are all represented on the tokens.  There is one token with a ‘1’ on it, two tokens with a ‘2’ on them, and so on, up until fifteen tokens with a ‘15’ on them.

A token is taken from the bag and the number on it is recorded, but it is not replaced in the bag.  A second token is then taken from the bag and the number on it is recorded.

Using a tree diagram, or otherwise, work out the probabilities of the following events:

(i) the numbers on the two tokens are neither both prime numbers nor both square numbers

(ii) the number on one of the tokens is a prime number, given that the numbers on the two tokens are neither both prime numbers nor both square numbers

(iii) the number on the first token is a prime number, given that the numbers on the two tokens are neither both prime numbers nor both square numbers

A crafty coyote spends most of his spare time trying to catch a very fast roadrunner bird.  The coyote’s schemes always involve one of three items procured from a well-known mail order retailer – a crate of TNT (with probability 0.51), a large boulder (with probability 0.29), or a rocket on wheels.

If the coyote uses a crate of TNT there is a 95% chance it will explode at the wrong time and injure the coyote while the roadrunner escapes; otherwise it will simply not explode at all and the roadrunner will escape.  If the coyote uses a large boulder there is an 85% chance it will injure him by landing on his head while the roadrunner escapes; otherwise it will injure the coyote by landing on his foot while the roadrunner escapes.  If the coyote uses a rocket on wheels, there is a 60% chance he will injure himself by running into a cliff face while the roadrunner escapes; otherwise the roadrunner will escape after tricking the coyote into riding the rocket off the top of a cliff.   If the coyote rides the rocket off the top of a cliff he will either get injured when the rocket explodes in mid-air, or get injured by crashing into a mountain on the other side of the valley, or land safely on the ground after his parachute unexpectedly functions properly.  It is twice as likely that the rocket will explode as it is that the coyote will crash into a mountain, and five times as likely that the coyote will crash into a mountain as it is that he will land safely using his parachute.

Draw a tree diagram to represent this information.

(i) Given that the coyote is injured during one of his schemes, find the exact probability that he is not injured by an explosion.

(ii) Given that the coyote is not injured during one of his schemes, find the exact probability that his scheme involved a rocket on wheels.

A wandering statistician informs the coyote that if he undertakes n of his schemes to capture the roadrunner, then he will have a better than 50% chance of not being injured during at least one of them.

 Find the smallest value of  that makes the statistician’s statement true.