Probability Diagrams : Tree & Venn Diagrams (Cambridge O Level Maths)

Exam Questions

4 hours54 questions
1
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3 marks

cie-igcse-2020-oct-nov-p4-tz3-q4a

Morgan picks two of these letters, at random, without replacement.

Find the probability that he picks

i)
the letter Y first,

[1]
ii)
the letter B then the letter Y.
 
[2]

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2a
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2 marks

On any Saturday, the probability that Arun plays football is 3 over 4.
On any Saturday, the probability that Bob plays football is 2 over 5.

Complete the tree diagram.

cie-igcse-2020-may-jun-p4-tz3-q7ai

2b
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2 marks

Calculate the probability that, one Saturday, Arun and Bob both play football.

2c
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3 marks

Calculate the probability that, one Saturday, either Arun plays football or Bob plays football, but not both.

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3
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3 marks

A soccer team plays two matches.
The tree diagram shows the probability of the team winning or losing the matches. 

cie-igcse-q22-tree-diagram

Find the probability that the soccer team wins at least one of the two matches.

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4
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2 marks

 box enclose space space space 1 space space space end enclose space space space space box enclose space space space 2 space space space end enclose space space space space box enclose space space space 3 space space space end enclose space space space space box enclose space space space 4 space space space end enclose space space space space box enclose space space space 5 space space space end enclose space 

The diagram shows five cards.
Two of the cards are taken at random, without replacement.

Find the probability that both cards show an even number.

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5a
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2 marks

calligraphic E equals{students in a school}
F equals{students who play football}
B equals{students who play baseball}
 
There are 240 students in the school.
 
• 120 students play football
• 40 students play baseball 
• 90 students play football but not baseball.

Complete the Venn diagram to show this information.
 

cie-igcse-2019-may-jun-p4-tz1-q6a 

5b
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1 mark

A student in the school is chosen at random.
 
Find the probability that this student plays baseball but not football.

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6
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2 marks

The Venn diagram below shows information about the number of gardeners who grow melons (M ), potatoes (P ) and carrots (C ).

cie-igcse-2019-may-jun-p2-tz2-q21b

A gardener is chosen at random from the gardeners who grow melons.
 
Find the probability that this gardener does not grow carrots.

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7
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2 marks

At a fitness centre, the number of members using the exercise machines (E), the swimming pool (S) and the tennis courts (T) is shown on the Venn diagram.

cie-igcse-2018-may-jun-1-p4-q10c

A member using the swimming pool is chosen at random.

Find the probability that this member also uses the tennis courts and the exercise machines.

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1
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3 marks

cie-igcse-2020-oct-nov-p4-tz3-q4a

Morgan picks two of these letters, at random, without replacement.

Find the probability that he picks two letters that are the same.

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2
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3 marks

A bag contains 4 red marbles and 2 yellow marbles.
Behnaz picks two marbles at random without replacement.

Find the probability that

i)

the marbles are both red,
 

[2]

ii)

the marbles are not both red.
 

[1]

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3a
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1 mark

40 children were asked if they have a computer or a phone or both.
The Venn diagram shows the results.

cie-igcse-2019-may-jun-p2-tz3-q20ai

A child is chosen at random from the children who have a computer.
 
Write down the probability that this child also has a phone.

3b
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2 marks

Complete the Venn diagram.
 

cie-igcse-2019-may-jun-p2-tz3-q20aii

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4a
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3 marks

The Venn diagram shows information about the number of elements in sets A, B and calligraphic E.
cie-igcse-2018-may-jun-2-p2-q23

straight n left parenthesis A union B right parenthesis equals 23

Find the value of x.

x equals................................................

4b
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2 marks

An element is chosen at random from calligraphic E.

Find the probability that this element is in left parenthesis A union B right parenthesis apostrophe.

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5a
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2 marks

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.

q19a-paper-1h-nov-2021-aqa-gcse-maths

5b
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3 marks

An item is chosen at random before the checks.

Work out the probability that the item is scrapped.

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1
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5 marks

cie-igcse-2020-oct-nov-p4-tz2-q6a

The diagram shows six discs.
Each disc has a colour and a number.

Two of the six discs are picked at random without replacement.

Find the probability that 

i)
both discs have the number 3,
  
[2]
ii)
both discs have the same colour.
 
[3]

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2a
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2 marks

Tanya plants some seeds.
The probability that a seed will produce flowers is 0.8.
When a seed produces flowers, the probability that the flowers are red is 0.6 and the probability that the flowers are yellow is 0.3.

Complete the tree diagram.

cie-igcse-2020-may-jun-p4-tz2-q7bi 

2b
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2 marks

Find the probability that a seed chosen at random produces red flowers.

2c
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3 marks

Tanya chooses a seed at random.
 
Find the probability that this seed does not produce red flowers and does not produce yellow flowers.

2d
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3 marks

Two of the seeds are chosen at random.
 
Find the probability that one produces flowers and one does not produce flowers.

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3a
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2 marks

Harris is taking a driving test.
The probability that he passes the driving test at the first attempt is 0.6.
If he fails, the probability that he passes at any further attempt is 0.75.

Calculate the probability that Harris passes the driving test at the second attempt.

3b
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2 marks

Calculate the probability that Harris takes no more than three attempts to pass the driving test.

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4
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4 marks

A bag contains 5 blue marbles and 2 green marbles.
Bryn picks one marble at random without replacement.
If this marble is not green, he picks another marble at random without replacement.
He continues until he picks a green marble.

Find the probability that he picks a green marble on his first, second or third attempt.

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5
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4 marks

Esme has a bag with 5 green counters and 4 red counters.
She takes three counters at random from the bag without replacement.
 
Work out the probability that the three counters are all the same colour.

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6a
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4 marks

cie-igcse-2018-oct-nov-p4-tz3-q7
 

Bag A contains 3 black balls and 2 white balls.
Bag B contains 1 black ball and 3 white balls.
A ball is taken at random from each bag.

i)
Show that a black ball is more likely to be taken from bag A than from bag B.
 
 

[1]

ii)
Find the probability that the two balls have different colours.
 
 

[3]

6b
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3 marks

The balls are returned to their original bags.
Three balls are taken at random from bag A, without replacement.

i)
Find the probability that they are all black.
 
 

[2]

ii)
Find the probability that they are all white.
 
 

[1]

6c
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3 marks

The balls are returned to their original bags.
A ball is taken at random from bag A and its colour is recorded.
This ball is then placed in bag B.
A ball is then taken at random from bag B.
 
Find the probability that the ball taken from bag B has a different colour to the ball taken from bag A.

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7a
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2 marks

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.

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7b
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4 marks

Is Anna more likely to win or to lose?

You must work out the probability that she wins.

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8a
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2 marks

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.

First jump Second jump Third jump

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8b
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2 marks

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

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1
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9 marks

cie-igcse-2020-oct-nov-p4-tz3-q4a

Morgan picks three of these letters, at random, without replacement.

Find the probability that

i)
all three letters are the same,
 
[2]
ii)
exactly two of the three letters are the same,
 
[5]
iii)
all three letters are different.
 
[2]

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2
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6 marks

Suleika has six cards numbered 1 to 6.

cie-igcse-2020-march-p4-tz2-q6a

Suleika takes two cards at random, without replacement.

i)
Find the probability that the sum of the numbers on the two cards is 5.


[3]

ii)
Find the probability that at least one of the numbers on the cards is a square number.


[3]

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3
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6 marks

cie-igcse-2019-oct-nov-p4-tz1-q8

The diagram shows 5 cards.

Donald chooses two of the five cards at random, without replacement.
He works out the total number of dots on these two cards.

i)
Find the probability that the total number of dots is 5.

[3]

ii)
Find the probability that the total number of dots is an odd number.

[3]

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4a
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1 mark

The probability that Andrei cycles to school is r.

Write down, in terms of r, the probability that Andrei does not cycle to school. 

4b
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8 marks

The probability that Benoit does not cycle to school is 1.3 minus r.
The probability that both Andrei and Benoit do not cycle to school is 0.4.

i)
Complete the equation in terms of r.
 
 left parenthesis......................... right parenthesis space cross times space left parenthesis......................... right parenthesis space equals space 0.4  
 
[1]
ii)
Show that this equation simplifies to 10 r squared minus 23 r plus 9 equals 0
 
[3]
iii)
Solve by factorisation 10 r squared minus 23 r plus 9 equals 0
 
r = ................... or r = ................... [3]
iv)
Find the probability that Benoit does not cycle to school. 
 
[1]

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5a
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2 marks

Angelo has a bag containing 3 white counters and x black counters.
He takes two counters at random from the bag, without replacement.

Complete the following statement.
 
The probability that Angelo takes two black counters is fraction numerator x over denominator x plus 3 end fraction cross times fraction numerator............ over denominator............ end fraction

5b
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8 marks

The probability that Angelo takes two black counters is 7 over 15.

i)
Show that 4 x squared minus 25 x minus 21 equals 0.
 
[4]
ii)
Solve by factorisation 4 x squared minus 25 x minus 21 equals 0.
 
x = .................... or x = ................. [3]
iii)
Write down the number of black counters in the bag.
 
[1]

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