Using Venn diagrams for conditional probability - Higher
Venn diagrams can be useful for organising information about frequencies and probabilities, which can then be used to solve conditional probability problems.
Example
- 90 pupils were asked whether they owned a laptop or a tablet device.
- 52 said they owned a laptop.
- 46 said they owned a tablet.
- 23 said they owned both.
Find the probability that a pupil chosen at random owns a laptop, given that they own exactly one device.
Image caption, 23 pupils answered that they owned a laptop and a tablet. As they own both, this number goes in the centre of the Venn diagram.
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The Venn diagram shows there are \(29 + 22 = 51\) pupils who own exactly one device (this becomes the denominator of the conditional probability). Out of these 51 pupils, 29 own a laptop. Therefore, the probability that a pupil chosen at random owns a laptop, given that they own exactly one device = \(\frac{29}{51}\).
Question
125 pupils were asked about their pets. 61 pupils said they had a cat and 68 pupils said they had a dog. 23 pupils said they had neither a cat nor a dog.
Show this information on a Venn diagram. Find the probability that a pupil chosen at random has a cat, given that they have a dog.
The total number of pupils with either a cat or dog is \(125 - 23 = 102\).
The total number of pupils with both a cat and a dog = \((61 + 68) - 102 = 27\).
The number of pupils with just a cat is \(61 - 27 = 34\).
The number of pupils with just a dog is \(68 - 27 = 41\).
This information is shown on the Venn diagram.
The Venn diagram shows there are \(27 + 41 = 68\) pupils who have a dog (this becomes the denominator of the conditional probability). Out of these 68 pupils, 27 have a cat. Therefore, the probability that a pupil chosen at random has a cat, given that they have a dog = \(\frac{27}{68}\).