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

and are three events with , ,  and 

Events and  are mutually exclusive. 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.

Complete the two-way table to show this information.

One student is chosen at random. 

Let represent the event that the student believes in unicorns.

Let represent the event that the student believes in vampires.

Find:

(i)

(ii)

(iii)

(iv)

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:

A student is chosen at random.

Let be the event that the student prefers ketchup.

Let be the event that the student is in middle school.

Find .

Determine whether the events and are independent. Give a reason for your answer.

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:

(i) enjoys watching exactly two of the three genres of film,

(ii) does not enjoy watching action musicals.

Three events  and , are such that and are mutually exclusive and and 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 and 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.

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.

Draw a two-way table to show this information.

One of the attendees is chosen at random.

Let be the event that the attendee likes anchovies.

Let be the event that the attendee likes bananas.

Find:

(i)

(ii)

(iii)

(iv)

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 .

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 .

,   and are three events with:

Events and are mutually exclusive. Events  and  are independent.

Draw a Venn diagram to illustrate the probabilities.

Find:

(i)

(ii)

(iii)

A group of people, aged 18 to 25, 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:

For the people in the sample, age and preferred holiday location are statistically independent.

Find the missing values in the table.

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)

• 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

• one quarter 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. 

Find the probability that the chess player enjoys playing:

(i) the Najdorf Sicilian and the Orangutan

(ii) exactly one of the three openings.

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

Given that the player enjoys playing at least two of the game openings, find the probability that the player enjoys playing the Ponziani countergambit.

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.

Find the probability that both marbles are different colours.

Given that both marbles are different colours, find the probability that the second marble is purple.

Rosco is a rural county sheriff who frequently finds himself involved in car chases. During any given car chase, Rosco runs into exactly one of three obstacles; a damaged bridge, an oil slick or a pig pen at the end of a dead-end road.

• There's a 47% chance that he runs into a damaged bridge.

• There's a 32% chance that he runs into an oil slick.

If he encounters a damaged bridge, there is a 25% chance that he makes 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 spins 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 pig pen at the end of a dead-end road, there is a 15% chance he stops his car in time; otherwise, he drives into the pig pen and ends up covered in mud.

Draw a tree diagram to represent this information.

Find the probability that, in a randomly chosen car chase, Rosco ends up covered in mud.

Given that Rosco ends up covered in mud in a randomly chosen car chase, find the probability that he did not encounter an oil slick.

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.

Find the probability that on that day Rosco does not end up covered in mud.

 Three events, ,  and , are such that:

• and  are independent

• and  are mutually exclusive

• 

• 

• 

• 

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

Find:

(i)

(ii)

(iii)

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.

, and  are three events such that 

In addition, the following two relations hold:

Draw a Venn diagram to illustrate the probabilities.

Find:

(i)

(ii)

(iii)

Events and  are such that    and  .

 Given that , find the range of possible values of  .

Additionally, event  is such that   and .

Find 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 

where    and  .

Find the following probabilities in terms of and :

(i)

(ii)

(iii)

A bag contains tokens. Each token has an integer from 1 to 15 written on it. For each integer in the interval , there are tokens with the number written on them. For example:

• There is one token with a ‘1’ on it,

• There are two tokens with a ‘2’ on them,

• There are 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.

Work out the probability that the numbers on the two tokens are neither both prime numbers nor both square numbers.

Work out the probability that 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

Work out the probability that 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 spinner is labelled with integers from 1 to 5. For each integer in the interval , the probability that the spinner lands on is , where is a constant.

Find the value of .

Given that the spinner lands on a prime number, find the probability that it lands on an even number.

The spinner is spun repeatedly. Find the minimum number of times it needs to be spun so that the probability that it lands on a 5 at least once is greater than 0.99.

A group of maths students and maths teachers are on a school trip. They have to choose between two activities: reading Latin poetry or partaking in extreme sports. The following incomplete two-way table shows part of the results of the survey:

For the people in sample, whether each person chooses Latin poetry or extreme sports is statistically independent of whether they are a student or a teacher.

Find the missing values and complete the table of survey results.

and  are two events with 

Given that  and  are independent, show that

In a school:

• 15% of students like cartoons but do not like love ballads

• 30% of student like love ballads but do not like cartoons

• Fewer than 25% of students do not like love ballads and do not like cartoons

Given that whether each student likes cartoons is independent of whether they like love ballads, find the percentage of students who like cartoons and love ballads.

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 ()

• the Assyrian king Ashurbanipal ()

• the Akkadian king Manishtushu ().

The following information is known.

• 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. 

• Being a fan of Ashurbanipal and being a fan of Manishtushu were independent events.

• 180 historians were polled in total. 

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 probability that the historian is not a fan of Suppiluliuma I.

A crafty coyote spends most of his spare time trying to catch a very fast roadrunner bird.  The coyote’s schemes always involve exactly one of three items:

• a crate of TNT (with probability 0.51)

• a large boulder (with probability 0.29)

• 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, the coyote is not injured, but the roadrunner still escapes.

If the coyote uses a large boulder, there is an 85% chance it will injure him by landing on his foot while the roadrunner escapes. Otherwise, the coyote is not injured, but the roadrunner still 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 coyote is not injured, but the roadrunner still escapes.

Draw a tree diagram to represent this information.

Given that the coyote is injured during one of his schemes, find the probability that his scheme involved a rocket on wheels.

A wandering statistician informs the coyote that if he tries of his schemes to capture the roadrunner, then he will have a better than 90% chance of not being injured during at least one of them.

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