Probability, sets, and events

Events have a set of (favorable) outcomes. We express sets with capital letters and elements with lower-case ones.

ex. roll 1d6:

• to get an even number is a set
• to get 2 is an event

Any set can be empty or have values in it. We denote the empty set: Ø

Non-empty sets can be finite or infinite depending on the number of elements.

• x ∈ A reads: “x belongs in set A” or “x in A”
• A ∋ x reads: “A contains X”
• ∉ not in
• ∨ for all/any
• : such that
• A sub-set is a set that is fully contained in another one. ex. A⊆B (A is a subset of B).
• Every set contains at least 2 subsets, itself and Ø

Multiple Events

Given sets of events A and B we can express each as a circle.

• Intersection: A∩B contains all the outcomes favorable for events A and B simultaneously.

  • If there are no elements the intersection is Ø.
  • If B⊆A ⇒ A∩B=B

• Unions: A∪B, we only require that either A or B happen ⇒ combination of all outcomes preferred for either A or B. Some observations:

• If A∩B = Ø ⇒ A∪B = A + B

• If A∩B != Ø ⇒ A∪B = A + B - A∩B

• If B⊆A ⇒ A∪B = A

• If B⊆A ⇒ If B happens A must happen, if A doesn't happen, B can't happen.

Mutually Exclusive Sets

Sets which are not allowed to have any overlapping elements:

• If A∩B = Ø ⇒ A∪B = A + B

The complement of a set contains all the values part of the sample space but not of the set. This means that the complement is != from mutually exclusive.

ex. even and ending in 5 are mutually exclusive. Not even (13) is not part of “ending in five”.

All complements are mutually exclusive but only some mutually exclusive sets are complements.