Partitioning, Counting Principles, and Basic Probability

Stats234 week 1 notes

Disjoint Events

Also known as mutually exclusive events

Events that are disjoint cannot occur at the same time, as a positive result from one, prevents a positive result in the other events.

For example: when tossing a coin once, you cannot get both heads and tails

Partitioning

Occurs for a sample space (S);

Is a division of S into a collection of events, either simple or compound, such that all of the events are mutually exclusive (disjoint) and mutually exhaustive

Both must be true for a valid partition:

• A∪A^c =S

• A∩A^c = ∅

Mutually Exhaustive

All of the events in a sample space add up exactly to the sample space, with no leftover space or events.

• A∪A^c =S

Mutually Exclusive

Same as Disjoint,

None of the events in a sample space can overlap.

• A∩A^c = ∅

What are the requirements for a partition of sample space (S) to be valid?

The events must be:

• disjoint (or mutually exclusive),

• mutually exhaustive,

• CANNOT be independent

Cardinality

The number of elements in the set

For some set, A, cardinality is denoted as N(A)

Fundamental Counting Principle

Also known as the multiplication rule

The number of ways in which a series of successive things can occur is found by multiplying the number of ways in which each thing can occur.

• Works best when there are no restrictions…ALL possible choices.

Example: If you own 5 pairs of jeans, 30 t-shirts, and 3 pairs of shoes, you have 5 × 30 × 3 = 450 possible outfits in your wardrobe

Permutation

The number of ways the arrange r out of n DISTINCT (no repeats) items in a row, keeping track of order

• read as “choosing r out of n,” with order taken into consideration

formula:

n!

(n - r)!

Combination

The number of ways to choose r out of n distinct items regardless of order

• Is read as “n choose r”

Formula:

n!

r!(n-r)!

Factorial

The number of ways to arrange n DISTINCT items in a row:

• Used when arranging or grouping objects

n!

Combinatorial Identities

De Morgan’s Laws

(A∪B)^c = A^c ∩ B^c

(A∩B)^c = A^c ∪ B^c

Equally Likely Events

The probability of an an event, A, in sample space, S, is equal to the cardinality of A (N(A)) divided by the cardinality of S (N(S)).

N(A)

N(S)

Conditional Probability

Given two events, A and B, we denote the probability of event A happening, given that event B is known to happen as: P(A|B)

• This reads as “A given B”

• The current information, or the certainty of event B, changes the sample space or the possible outcomes

General Addition Rule

P(A∪B) = P(A) + P(B) - P(A∩B)

The formula above only applies to disjoint event. For independent events,

P(A∪B) = P(A) + P(B)

General Multiplication Rule

P(A∩B) = P(A) x P(B|A)

Probability Axioms

• Axiom 1: For any event, A, P(A) > 0

• Axiom 2: P(S) = 1, where S = Sample Space

• Axiom 3: If A₁, A₂, A₃, … is an infinite collection of disjoint events, then:

  • P(A₁∪A₂∪A₃∪…) = P(A₁) + P(A₂) + P(A₃) + …

Compliment Rule

P(A^c) = 1 - P(A)

Independence Checks

Two events A and B are said to be independent if any of the following are true:

• P(A|B) = P(A)

  • B happening does not affect A happening

• P(A|B) = P(A|B^c)

  • How likely A is to happening is not affected by B happening or not

• P(B|A) = P(B)

  • A happening doesn’t affect how likely B happens

• P(A∩B) = P(A) x P(B)

If any are true, then all are true

Multiplication Rule for Independent Events

P(A∩B) = P(A) x P(B)

Multinomial Arrangement Formula

Is used to find the number of unique arrangements when arranging objects that have repetitions:

If you have:

• n total objects

• with repeats of sizes n₁, n₂, n₃, etc.

When a problem that deals with counting asks about arranging objects, use:

factorials

How to prove if an event is disjoint?

P(A∩B) = ∅

Or, in other words, there can be no intersection between the events.

Can a Disjoint event be an Independent Event?

No, disjoint events cannot be an independent event and an independent event cannot be disjoint.

What are the similarities and differences between combinations and permutations?

Similarities:

• Both are used for selecting groups

• Both deal with distinct, or non repeating, items

Differences

• Slightly different formula

• order DOES NOT matter for combinations

• order MATTERS for permutations

Can you do combinations with replacement of objects?

Yes! This is known as a combination with repetition or a multiset selection

• For this type of combination, order does not matter, and items can be repeated.

• The formula is different:

(n+r-1)!

r!(n-1)!

General Multiplication Rule for several events

P(A∩B∩C) = P(ABC) = P(A) x P(B|A) x P(C|AB)

P(A∩B∩C∩D) = P(ABCD) = P(A) x P(B|A) x P(C|AB) x P(D|ABC)

etc