KS3

How to find probabilities from Venn diagrams

Part ofMathsSets and Venn diagrams

Key points

An image of a Venn diagram. The Venn diagram has been used to classify the location where animals live. One circle represents, the north pole, the other circle represents, the south pole. — Image caption,
Venn diagrams are used to sort sets of data.

Venn diagrams are often used to find the probability of events. They are used to sort data into sets, which may be presented showing all the individual or showing the number of elements in each . The notation for ‘the number of elements in set \(A\)' is \(n(A)\).
To calculate the probability of an event the number of elements in the relevant region and the total number of elements in the , \(n(ξ)\), must be known.
The probability of an event not happening uses the complement of the region on the Venn diagram.

An image of a Venn diagram. The Venn diagram has been used to classify the location where animals live. One circle represents, the north pole, the other circle represents, the south pole. — Image caption,
Venn diagrams are used to sort sets of data.

Calculating probabilities from a Venn diagram

For a that shows all the elements in the Venn diagram:
- Count the number of elements in each region.
- Redraw the Venn diagram, writing the number of elements instead of individual elements.
To calculate the of an event:
- Find the number of elements in the region that represents the event, this is \(n\)(event).
- Find the total number of elements, this is \(n(ξ)\).
- Divide the number of elements of the event by the total number of elements.

Examples

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A group of animals are sorted into where they may naturally be found. Work out the probability that one of the animals, chosen at random, lives in the Antarctic (South Pole).
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Instead of showing each individual named element, as here, the Venn diagram can be redrawn, by replacing the animals in each region of the diagram with numbers.
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This is done by counting the elements in each region of the Venn diagram, replacing the named elements with the number of elements in each region.
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The event ‘lives at the South Pole’ is the region inside the circle for the South Pole. The number of animals at the South Pole is 3 + 2, 𝒏(South Pole) = 5. The total number of animals is 5 + 3 + 2 + 13, 𝒏(ξ) = 13. These two values are used to calculate the probability.
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P(South Pole) is the number of animals at the South Pole divided by the total number of animals in the whole group (the universal set). This is 𝒏(South Pole) ÷ 𝒏(ξ). 5 ÷ 13 = ⁵⁄₁₃.
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The integers from 21 to 30 are sorted by whether they are multiples of 2 and multiples of 3. Find the probability that a number chosen at random is a multiple of both 2 and 3
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Count the number of elements in each region and redraw the Venn diagram to show this.
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The probability that the number is a multiple of both 2 and 3 uses the intersection of the set of multiples of 2 and the set of multiples of 3. There are two elements in this region. The total number of elements is 3 + 2 + 2 + 3 = 10
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P(multiple of 2 and multiple of 3) is the number of elements that are multiples of both 2 and 3 divided by the total number of elements. 2 ÷ 10 = 0۰2 or ⅕.

Question

The Venn diagram shows the result of some people saying whether they like tea and coffee.

Find the probability that a person chosen at random likes either tea or coffee but not both.

An image of a Venn diagram with two intersecting circles. The circle on the left, is labelled, Tea. The circle on the right, is labelled, Coffee. A rectangle has been drawn around the outside of the two circles. Outside the rectangle, in the top left: the symbol, ξ. The Venn diagram has been populated with numbers. The number, seventeen, is in the intersection. The number, eighteen, is in the other part of the circle labelled tea. The number, ten, is in the other part of the circle labelled coffee. The number, five, is outside the circles, in the rectangular box.

Calculating probabilities from Venn diagrams and using set notation

Probabilities using set :
- P(\(A\)) is the probability of the event represented by set \(A\).
- P(\(A'\)) = P(not \(A\)). \(A’\) is the of \(A\).
- P(\(A\)∩\(B\)) = P(\(A\) and \(B\)). \(A\)∩\(B\) is the intersection of set \(A\) and set \(B\).
- P(\(A\)ᴜ\(B\)) = P(\(A\) or \(B\) or both). \(A\)ᴜ\(B\) is the union of set \(A\) and set \(B\).
To calculate the probability of an event:
- Find the number of elements in the Venn diagram, this is \(n\)(\(ξ\)).
- P(\(A\)) = \(n\)(\(A\)) ÷ \(n\)(\(ξ\))
- The answer may be written as a fraction, decimal or percentage.

Examples

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The Venn diagram shows two holiday destinations that people have travelled to, 𝑺 = {Spain} and 𝑮 = {Greece}. Find the following probabilities. P(𝑮), P(𝑺’), P(𝑺∩𝑮) and P(𝑺ᴜ𝑮).
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First find the total number of people in the Venn diagram. This is 𝒏(ξ). Add up all the numbers in the Venn diagram. 5 + 30 +10 + 15 = 60. The number of people in the survey is 60
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P(𝑮) is the probability that a person selected at random has been to Greece. P(𝑮) = 𝒏(𝑮) ÷ 𝒏(ξ). The number of people who have travelled to Greece, 𝒏(𝑮) is 30 + 10 = 40. The total number of people, 𝒏(ξ) is 60. 40 ÷ 60 = ⁴⁰⁄₆₀, this fraction can be simplified. P(𝑮) = ⅔ . The probability that a person has been to Greece is ⅔.
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P(𝑺’) is the probability that a person has not been to Spain. 𝑺’ is the complement of set 𝑺, the region that is outside the circle for set 𝑺. 10 + 15 = 25. P(𝑺’) = 𝒏(𝑺’) ÷ 𝒏(ξ). 25 ÷ 60 = ²⁵⁄₆₀, this simplifies to ⁵⁄₁₂. The probability that a person has not been to Spain is ⁵⁄₁₂.
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P(𝑺∩𝑮) is the probability that a person has been to both Spain and Greece. 𝑺∩𝑮 is the intersection of set 𝑺 and set 𝑮, this is the region that is the overlap of the two circles. The number of elements in the intersection, 𝒏(𝑺∩𝑮) is 30. P(𝑺∩𝑮) = 𝒏(𝑺∩𝑮) ÷ 𝒏(ξ). 30 ÷ 60 = ³⁰⁄₆₀, this simplifies to ½. The probability that a person has been to both Spain and Greece is ½ or 0۰5
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P(𝑺ᴜ𝑮) is the probability that a person has been to either Spain or Greece, or both. 𝑺ᴜ𝑮 is the union of set 𝑺 and set 𝑮, this is the region that is the combination of the two circles. The number of elements in the union, 𝒏(𝑺ᴜ𝑮) is 5 + 30 + 10 = 45. P(𝑺ᴜ𝑮) = 𝒏(𝑺ᴜ𝑮) ÷ 𝒏(ξ). 45 ÷ 60 = ⁴⁵⁄₆₀, this simplifies to ¾. The probability that a person has been to either Spain or Greece or both is ¾ or 0۰75
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The Venn diagram, showing the gifts that eleven people re-gift, is incomplete. 𝑨 = {scarves} and 𝑩 = {candles}. 𝒏(ξ) = 11. Find P(𝑨).
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𝒏(ξ) = 11, ? + 1 + 3 + 5 = 11, which means that the missing value ? = 2
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P(𝑨) = 𝒏(𝑨) ÷ 𝒏(ξ). n(𝑨) is the number of people in set 𝑨, this is 2 + 1 = 3. P(𝑨) = ³⁄₁₁.

Question

The Venn diagram shows items people donate to charity.

\(A\) = {clothing} and \(B\) = {books}.

Find P(\(B’\)).

An image of a Venn diagram with two intersecting circles. The circle on the left, is labelled, A. The circle on the right, is labelled, B. A rectangle has been drawn around the outside of the two circles. Outside the rectangle, in the top left: the symbol, ξ. The Venn diagram has been populated with numbers. The number, sixty two, is in the intersection. The number, thirty three, is in the other part of the circle labelled A. The number, eighteen, is in the other part of the circle labelled B. The number, eighty seven, is outside the circles, in the rectangular box.

Practise working out finding probabilities from Venn diagrams

Practise working out finding probabilities from Venn diagrams with this quiz. You may need a pen and paper to help you with your answers.

Quiz

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