Conditional Probability in Venn Diagrams

Slideshow 59, MathematicsMr 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 sections. (including the outside).

16

4 Set Venn DiagramsSome 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 Set B

𝐴∩𝐵 ′𝐴∩𝐵𝐴 ′ ∩𝐵

𝐴 ′ ∩𝐵 ′

The total number of values in this diagram is the number of entries surveyed.

Number of Entries =𝐴∩𝐵′+ 𝐴∩𝐵+𝐴′∩𝐵+ 𝐴 ′ ∩𝐵 ′𝑃 (𝐴∩𝐵 ′)+𝑃 ( 𝐴∩𝐵 )+𝑃 (𝐴′∩𝐵 )+𝑃 ( 𝐴′∩𝐵′ )=¿1

ProbabilitiesIf we choose one of the entries surveyed at random, we can find the probability of it being in its given category.

Set A Set B

1371911

ExampleTotal # of entries: 50

𝑃 ( 𝐴)=¿2050𝑃 (𝐵 ′ )=¿2450𝑃 (𝐴∩𝐵)=¿750

𝑃 (𝐴∩𝐵 ′ )=¿1350𝑃 (𝐴 ′ ∩𝐵 ′)=¿1150𝑃 (𝐴∪𝐵)=¿3950Note: . 𝑃 (𝐴 ′ ∩𝐵 ′)

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 hours

Being able to kiss a girl after eating garlic

Conditional Probability

For this we use given that which has the ‘|’ symbol. What does mean?It means, assuming A is true, find the chance of B being true.

Set A Set B2435

𝑃 (𝐵|𝐴 )=¿37which is different from . 814

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.

How about for the Venn diagram on the right?

Conditional Probability

Set A Set B

914423

Example

Total # of entries: 50

𝑃 ( 𝐴)=¿2350𝑃 (𝐵∨𝐴 )=¿1423𝑃 ( 𝐴∨𝐵 )=¿1418𝑃 ( 𝐴∨𝐵 ′ )=¿932

Try the ActivExpression exercise!

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.

If picked randomly, find…

6401440526513813

Set A Set B

29301328

100𝑃 (𝐴∩𝐵 ′)= 30

100

𝑃 (𝐵∨𝐴 )=2959

(A)

(A)

Set A Set B

Set C