Probability Diagrams - Venn & Tree Diagrams

50 people were asked if they speak French or German or Spanish.

Of these people,

31 speak French
2 speak French, German and Spanish
4 speak French and Spanish but not German
7 speak German and Spanish
8 do not speak any of the languages
all 10 people who speak German speak at least one other language

Two of the 50 people are chosen at random.

Work out the probability that they both only speak Spanish.

Finlay plays two tennis matches.

The probability that he will win a match and the probability that he will lose a match are shown in the probability tree diagram.

Work out the probability that Finlay wins both matches.

Work out the probability that Finlay loses at least one match.

Louise makes a spinner.

  The spinner can land on green or on red.
  The probability that the spinner will land on green is 0.7

  Louise spins the spinner twice.

  Complete the probability tree diagram.

Work out the probability that the spinner lands on two different colours.

Machine A and machine B make bottles.

The probability that a bottle made by machine A is faulty is 0.02
The probability that a bottle made by machine B is faulty is 0.05

Complete the probability tree diagram.

Shazia takes at random one bottle made by machine A and one bottle made by machine B.

Work out the probability that at least one of these bottles is faulty.

Mary travels to work by train every day.
The probability that her train will be late on any day is 0.15

Complete the probability tree diagram for Thursday and Friday.

Work out the probability that her train will be late on at least one of these two days.

A darts team is going to play a match on Saturday and on Sunday.
The probability that the team will win on Saturday is 0.45

If they win on Saturday, the probability that they will win on Sunday is 0.67
If they do not win on Saturday, the probability that they will win on Sunday is 0.35

Complete the probability tree diagram.

Find the probability that the team will win exactly one of the two matches.

Yvonne has 10 tulip bulbs in a bag.

7 of the tulip bulbs will grow into red tulips.
3 of the tulip bulbs will grow into yellow tulips.

Yvonne takes at random two tulip bulbs from the bag.
She plants the bulbs.

Complete the probability tree diagram.

Work out the probability that at least one of the bulbs will grow into a yellow tulip.

and are two sets.


Complete the Venn diagram to show the number of elements in each region.

Find

i)

[1]

ii)

[1]

One of the elements from the universal set is chosen at random.

Find the probability that this element is in set .

There are 31 students in a class.

The only languages available for the class to study are French and Spanish.
17 students study French.
15 students study Spanish.
6 students study neither French not Spanish.

Using a Venn diagram, or otherwise, work out how many students study only one language.

One of the students is chosen at random.

Find the probability that this student studied French but not Spanish.

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}
A = {3, 7, 11, 13}
B = {3, 6, 9, 12, 13}
C = {2, 3, 5, 6, 7, 8}

Complete the Venn diagram

List the members of the set B' ∩ C.

List the members of the set (A U C)' ∩ B.

Find n(A' ∩ B')

One of the numbers in the Venn diagram is chosen at random.

Find the probability that this number is in set (A ∩ B)

Each student in a group of 32 students was asked the following question.

"Do you have a desktop computer (D), a laptop (L) or a tablet (T)?"

Their answers showed that,

19 students have a desktop computer,
17 students have a laptop,
16 students have a tablet,
9 students have both a desktop computer and a laptop,
11 students have both a desktop computer and a tablet,
7 students have both a laptop and a tablet,
5 students have all three.

Using this information, complete the Venn diagram to show the number of students in each appropriate subset.

One of the students with both a desktop computer and a laptop is chosen at random.

Find the probability that this student also has a tablet.

There are 20 glasses in a cupboard.

   13 of the glasses are large
     7 of the glasses are small

Roberto takes at random two glasses from the cupboard.

Items made at a factory have to pass two checks.

90% pass the first check.
The items that fail are scrapped.
99% of the items that pass the first check pass the second check.
The items that fail are scrapped.

Complete the tree diagram.

An item is chosen at random before the checks.

Work out the probability that the item is scrapped.

Two ordinary fair dice are rolled.

Complete the tree diagram.

Work out the probability that both dice land on a number less than 3

Work out the probability that exactly one of the dice lands on a number less than 3

The probability that any postcard posted in Portugal on Monday is delivered to the UK within a week is 0.62.
The probability that any postcard posted in Portugal on Friday is delivered to the UK within a week is 0.41.

Sergio is in Portugal.
He posts one postcard to the UK on Monday.
He posts another postcard to the UK on Friday.

The Venn diagram shows some information about 150 students.

 = 150 students
 = students who study Chemistry
 = students who study Physics

The probability that a Physics student, chosen at random, also studies Chemistry is  

One of the 150 students is chosen at random.

Work out the probability that the student does not study either Chemistry or Physics.

ξ = 29 students in a class

C = students who own a cat

D = students who own a dog

A student is chosen at random.

Circle the probability that the student owns a cat or a dog but not both.

A student who owns a dog is chosen at random.

Circle the probability that the student also owns a cat.

The Venn diagram shows information about some houses.

G = houses with a garage
S = houses with a shed

A house is chosen at random.

The house has a garage.

What is the probability that it has a shed?

The house does not have a garage.

What is the probability that it does not have a shed?

Show that      

In the Venn diagram

   ξ represents 31 students in a class
   C is students who have a cat
   D is students who have a dog

One student from the class is picked at random.

Work out the probability that the student has a dog.

One of the students who has a cat is picked at random.

Work out the probability that this student has a dog.

A school has 86 teachers.

42 are male and 44 are female.

of the male teachers have blue eyes.

of the female teachers have blue eyes.

ξ   = teachers in the school
M = male teachers
B  = teachers who have blue eyes

Complete the Venn diagram.

One teacher who has blue eyes is chosen at random.

Work out the probability that the teacher is male.

The Venn diagram shows the number of students studying Mathematics (M) and the number of students studying Physics (P) in a college.
35 students do not study either subject.

The total number of students is 121.
Find the value of .

= .................... 

One of the 121 students is selected at random.

Find the probability that this student studies Mathematics, given that they study Physics.

A skills test has two sections, literacy (L) and numeracy (N).
One day everyone who took the skills test passed at least one section.
88% passed the literacy section and 76% passed the numeracy section.

Represent this information on a Venn diagram.
Show clearly the percentage in each section of the diagram.

One person is chosen at random from all the people who took the skills test that day.

What is the probability that this person

[2]

[2]

50 people attended an outdoor activity day.

• 40 took part in walking.
• 18 took part in sailing.
• 3 did neither activity.

One of the people who walked is chosen at random.

Find the probability that this person also sailed.

Here are the results of a survey of 437 people in a town.

• 62 males speak Spanish.
• 153 females do not speak Spanish.
• 280 people do not speak Spanish.

Jeff says

   At least 2 out of every 5 females in the town can speak Spanish.

Is he correct?
Show clearly how you reached your decision.

In a group of 120 adults, 85 watch football, 78 play a sport and 20 do neither.

Find the probability that an adult chosen at random from those who watch football does not play a sport.

Here is a Venn diagram.

Write down the numbers that are in set

i) 

[1]

ii) 

[1]

One of the numbers in the diagram is chosen at random.

Find the probability that the number is in set .

There are 30 students in Mr Lear's class.
16 of the students are boys.

Two students from the class are chosen at random.

Mr Lear draws this probability tree diagram for this information.

Write down one thing that is wrong with the probabilities in the probability tree diagram.

Owen and Wasim play for the school football team.

The probability that Owen will score a goal in the next match is 
The probability that Wasim will score a goal in the next match is 

Mr Slater says,

"The probability that both boys will score a goal in the next match is "

Is Mr Slater right?
Give a reason for your answer.

Wendy goes to a fun fair.
She has one go at Hoopla.
She has one go on the Coconut shy.

The probability that she wins at Hoopla is 0.4
The probability that she wins on the Coconut shy is 0.3

Complete the probability tree diagram.

Work out the probability that Wendy wins at Hoopla and also wins on the Coconut shy.

When a biased 6-sided dice is thrown once, the probability that it will land on 4 is 0.65
The biased dice is thrown twice.

Amir draws this probability tree diagram.
The diagram is not correct.

Write down two things that are wrong with the probability tree diagram.

The probability that it will rain on a day in June is 0.2

When it rains the probability that my tennis match is cancelled is 0.7

When it does not rain, the probability that my tennis match is not cancelled is 0.95

Complete the probability tree diagram for this information.

Work out the probability that, on a day in June, it does not rain and my tennis match is cancelled.

The numbers from to  are shown in the Venn diagram.

List the members of the set

[1]

List the members of the set  

[1]

A number is picked at random from the numbers in the Venn diagram.

Find the probability that this number is in set but is not in set .

Two events and are such that  ,    and 

Given that  ,

complete the Venn diagram to show the number of elements in each region.

An element is chosen at random from .

Using the Venn diagram, find the probability that this element is in

Aika has 2 packets of seeds, packet A and packet B

There are 12 seeds in packet A and 7 of these are sunflower seeds.
There are 15 seeds in packet B and 8 of these are sunflower seeds.
 
Aika is going to take at random a seed from packet A and a seed from packet B

Complete the probability tree diagram.

Calculate the probability that Aika will take two sunflower seeds.

Harry has two fair 5-sided spinners.

Harry is going to spin each spinner once.

Complete the probability tree diagram.

Work out the probability that at least one of the spinners will land on green.

A fair spinner has five equal sections numbered 1, 2, 3, 4 and 5

A fair six-sided dice has five red faces and one green face.

The spinner is spun.

If the spinner shows an even number, the dice is thrown.

Complete the tree diagram for the spinner and the dice.

Work out the probability of getting an even number and the colour green.

A weather forecast says

• the probability that it will rain on Monday is 0.55
  and
• the probability that it will rain on Tuesday is 0.25.

Ella draws a tree diagram to show this information.

Write down three errors that Ella has made with her tree diagram.

A bag contains 4 red counters and 3 blue counters only.
Jack picks a counter at random and then replaces it.
Jack then picks a second counter at random.

Complete the tree diagram.

Work out the probability that Jack picks two red counters.

Rashid drives his car along a road passing through two sets of traffic lights.

The tree diagram shows the probabilities of the lights being red when he reaches them.

Complete the tree diagram.

Write down the probability that the first set is not red.

Given that the first set is red, write down the probability that the second set is not red.

Work out the probability that both sets are not red.

Work out the probability that at least one set is not red.

A and B are two events.

Some probabilities are shown on the Venn diagram.

Work out 

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

S = square numbers

E = even numbers

Complete the Venn diagram.

One of the numbers is chosen at random.

Write down

In a survey, 60 students were asked whether they have a bank account (B) and whether they have a part-time job (J).

The number of students who had neither a bank account nor a part-time job was x.

The Venn diagram shows the results in terms of x.

One of the 60 students is chosen at random.

Find the probability that they have a bank account.
Show your working.

A and B are two sets of traffic lights on a road.

The probability that a car is stopped by lights A is 0.4

If a car is stopped by lights A, then the probability that the car is not stopped by lights B is 0.7

If a car is not stopped by lights A, then the probability that the car is not stopped by lights B is 0.2

Complete the probability tree diagram for this information.

Mark drove along this road.
He was stopped by just one of the sets of traffic lights.

Is it more likely that he was stopped by lights A or by lights B?
You must show your working.

Cody has two bags of counters, bag and bag .

Each of the counters has either an odd number or an even number written on it.

There are 10 counters in bag and 7 of these counters have an odd number written on them.
There are 12 counters in bag and 7 of these counters have an odd number written on them.

Cody is going to take at random a counter from bag  and a counter from bag .

Complete the probability tree diagram.

Calculate the probability that the total of the numbers on the two counters will be an odd number.

Harriet also has a bag of counters.
Each of her counters also has either an odd number or an even number written on it.

Harriet is going to take at random a counter from her bag of counters.

The probability that the number on each of Cody’s two counters and the number on Harriet’s counter will all be even is 

Find the least number of counters that Harriet has in her bag.
Show your working clearly.

Each day that Barney goes to college, he either goes by bus or he walks.
The probability that Barney will go to college by bus on any day is 0.3

When Barney goes to college by bus, the probability that he will be late is 0.2
When Barney walks to college, the probability that he will be late is 0.1

Complete the probability tree diagram.

Barney will go to college on 200 days next year.

Work out an estimate for the number of days Barney will be late for college next year.

There are 32 students in a class.
In one term these 32 students each took a test in Maths , in English and in French .

25 students passed the test in Maths.
20 students passed the test in English.
14 students passed the test in French.
18 students passed the tests in Maths and English.
11 students passed the tests in Maths and French.
4 students failed all three tests.
students passed all three tests.

The incomplete Venn diagram gives some more information about the results of the 32 students.

Use all the given information about the results of students who passed the test in Maths to find the value of .

....................................................... 

Use your value of to complete the Venn diagram to show the number of students in each subset.

A student who passed the test in Maths is chosen at random.

Find the probability that this student failed the test in French.

Felix has 10 cards.
There are 5 red cards, 4 blue cards and 1 green card.

Felix takes at random one of the cards.
He does not replace the card.
Felix then takes at random a second card.

Complete the probability tree diagram.

Work out the probability that Felix takes at least one blue card and no green card.

Magnus and Garry play 2 games of chess against each other.

The probability that Magnus beats Garry in any game is 

The probability that any game between Magnus and Garry is drawn is 

The result of any game is independent of the result of any other game.

Complete the probability tree diagram.

For each game of chess,

the winner gets 2 points and the loser gets 0 points,

when the game is drawn, each player gets 1 point.

Work out the probability that, after 2 games, Magnus and Garry have the same number of points.

Emilie takes part in two races.

The probability that she wins the first race is 0.7
The probability that she wins the second race is 0.4
The outcomes of the two races are independent.

Complete the probability tree diagram.

Work out the probability that Emilie wins exactly one of the two races.

Emilie is going to take part in a third race.

If she wins both of the first two races, the probability that she will win the third race is 0.6

If she wins exactly one of the first two races, the probability that she will win the third race is 0.3

Work out the probability that Emilie will win exactly two of the three races.

In group , there are 6 girls and 8 boys.
In group , there are 3 girls and 7 boys.

A team is made by picking at random one child from group  and one child from group .

Complete the probability tree diagram.

Work out the probability that there are two boys in the team.

After the first team has been picked, a second team is picked.
One child is picked at random from the children left in group  and one child is picked at random from the children left in group   .

Work out the probability that there are two boys in each of the two teams.

Anna plays a game with an ordinary, fair dice.

If she rolls 1 she wins.
If she rolls 2 or 3 she loses.
If she rolls 4, 5 or 6 she rolls again.

When she has to roll again,

if she rolls an odd number she wins
if she rolls an even number she loses.

Complete the tree diagram with the four missing probabilities.

Is Anna more likely to win or to lose?

You must work out the probability that she wins.

On Friday, Greg takes part in a long jump competition.
He has to jump at least 7.5 metres to qualify for the final on Saturday.

• He has up to three jumps to qualify.
• If he jumps at least 7.5 metres he does not jump again on Friday.

Each time Greg jumps, the probability he jumps at least 7.5 metres is 0.8
Assume each jump is independent.

Complete the tree diagram.

Work out the probability that he does not need the third jump to qualify.

Reuben is playing a matching game with these cards.

He turns the cards over and shuffles them.
Reuben takes a card and keeps it. He then takes a second card.
If the cards are different, he wins the game.

Complete this tree diagram to show the probabilities for each card picked in the game.

What is the probability that Reuben wins the game?