April 14, 2024

Single events (Venn diagrams)

Probability for single events can be visually expressed through Venn diagram

single event

Sample space the list of all possible outcomes.

Event the outcomes that meet the requirement.

Probability for event A,

probability

Here the shaded circle.

Imagine I have a fruit bowl containing 10 pieces of fruit: 6 apples and 4 bananas.
single events
I pick a piece of fruit. Below are some common situations with Venn diagrams.

Note: These events are also exhaustive as there is nothing outside of the events (nothing in the sample space).

Mutually exclusive

Example: What is the probability of picking each fruit?

Events do not overlap

P(A∪B) = P(A) + P(B)

P(A∩B) = 0

Mutually exclusive
Mutually exclusive
Note: In independent events P(A∩B) = P(A) × P(B). It will often be stated in questions if events are independent.

Combined events

Example: Of the apples 2 are red, 2 are green and 2 are yellow. What is the probability of picking a yellow apple?

The intersect is the area the events overlap.

P(A∩B) = P(A) + P(B) − P(A∪B)

Combined events
Combined events

Example: What is the probability of picking an apple or a yellow fruit? 

The union is the area contain by both events.

P(A∪B) = P(A) + P(B) − P(A∩B)

When an event is exhaustive the probability of the union is 1.

Combined events

A: apples

B: yellow fruit

Event is exhaustive so probability of union is 1.

Compliment

Example: What is the probability of not picking a yellow fruit? 

Everything that is not in the stated event.

P(A′) = 1 − P(A)

Compliment

A: apples

B: yellow fruit

P(B′) = 1 − P(B) = 1 − 0.6 = 0.4

Conditional

Example: What is the probability of picking an apple given I pick a yellow fruit?

 The probability given that some condition is already in place.

Conditional
Conditional
Conditional
You can think of this as using B as the sample space, or removing the non yellow apples from the fruit bowl before choosing.
Conditional