Conditional Probability
in Venn Diagrams
Slideshow 59, Mathematics
Mr Richard Sasaki, Room 307
Objectives
• Learn to draw Venn Diagrams for Four Sets
• Review and picture sections in simple Venn
Diagrams (two sets)
• Understand Conditional Probability in Venn
Diagrams
4 Set Venn Diagrams
A regular Venn Diagram must cover all options.
It cannot be made simply with just circles.
This pattern does not cover
Red and Green or Yellow and
Blue.
In fact, to cover all possibilities,
there must be 16 sections.
(including the outside).
4 Set Venn Diagrams
Some solutions are…
This one has 11 sets… (It was discovered 3 years ago.)
What makes this special is
that it’s symmetrical.
Introduction
We know how to use Venn Diagrams now (this is
our third slideshow on Venn Diagrams).
Set A
𝐴 ∩ 𝐵′
Set B
𝐴∩𝐵
𝐴′ ∩ 𝐵
The total number of values
in this diagram is the
number of entries surveyed.
𝐴′ ∩ 𝐵′
Number of Entries = 𝐴 ∩ 𝐵′ + 𝐴 ∩ 𝐵 + 𝐴′ ∩ 𝐵 + 𝐴′ ∩ 𝐵′
𝑃 𝐴 ∩ 𝐵′ + 𝑃 𝐴 ∩ 𝐵 + 𝑃 𝐴′ ∩ 𝐵 + 𝑃 𝐴′ ∩ 𝐵′ = 1
Probabilities
If we choose one of the entries surveyed at random,
we can find the probability of it being in its given
category.
Example
11
Set A
13
Set B
7
19
Total # of entries: 50
20
𝑃 𝐴 = 50
24
𝑃 𝐵′ = 50
𝑃(𝐴 ∩ 𝐵) =
𝑃(𝐴 ∩ 𝐵′) =
13
50
𝑃(𝐴′ ∩ 𝐵′) =
11
50
7
50
𝑃(𝐴 ∪ 𝐵) =
Note: 𝑃 𝐴 ∪ 𝐵 = 1 − 𝑃(𝐴′ ∩ 𝐵′).
39
50
Shading Exercise
Firstly I’d like us to do a shading in exercise. Shade
in the areas mentioned on your worksheet.
Answers
Conditional Probability
Conditional probability refers to…
The likelihood of something happening changing,
depending on whether something else happens
or not. (For example: If A happens, P(B) is lower
than when A doesn’t happen.)
When in daily life might this happen?
Wearing Glasses and taking an eye
exam
Passing an exam after studying for 𝑥
hours
Being able to kiss a girl after eating
garlic
Conditional Probability
Note: Most of your examples would have been
two events happening in turn (like with tree
diagrams). But Venn diagrams show relationships
as a whole, ignoring time.
For this we use given that which has the ‘|’
symbol. What does P(𝐵|𝐴) mean?
It means, assuming A is true,
find the chance of B being
2 Set A Set B
true.
4 3 5
How about for the Venn
diagram on the right?
𝑃 𝐵 𝐴 = 3 7 which is different from 𝑃 𝐵 =8 14.
Conditional Probability
Example
A group of 50 people are asked whether they wear glasses
and whether they wear contact lenses. 23 people wear
glasses (Set A), 18 wear contacts (Set B) and 14 use both.
23
Set A
9
Set B
14
4
Total # of entries: 50
If picked randomly, find…
23
𝑃 𝐴 = 50
𝑃 𝐵|𝐴 =
𝑃 𝐴|𝐵 =
𝑃 𝐴|𝐵′ =
Try the ActivExpression exercise!
14
23
14
18
9
32
6
14 40
40 5
5 26
13 8
13
100
30
𝑃 𝐴∩𝐵 =
100
29
𝑃 𝐵|𝐴 =
59
′
𝑃(A ∪ 𝐵)=
𝑃(A′|𝐵′)=
Set A
Set B
30 29 13
72
100
28
58
28
Set A
Set C
Set B