# Probability

#### Subjective Interpretation

• P(Impossibility) = 0
• P(Win the Lottery) ≈ 0
• P(Sure Thing) = 1
• P(Death and Taxes) ≈ 1 (basically assured)
• P(“Heads on a coin toss”) = 0.5
• P(Gender of newborn is Female) ≈ 0.5

#### Overcounting

• It is when you are doing unions and you count the same data more than once
• Happens when two data sets have overlap
• To counter for this, with two data sets
• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• To counter for this, with three data sets
• P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

### Laws/ Identities

#### Commutative Laws

• A ∪ B = B ∪ A
• A ∩ B = B ∩ A

#### Associative Laws

• A ∪ B ∪ C = (A ∪ B) ∪ C = A ∪ (B ∪ C)
• A ∩ B ∩ C = (A ∩ B) ∩ C = A ∩ (B ∩ C)

#### Distributive Laws

• (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
• (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)

#### DeMorgan’s Laws

• (A ∩ B)’ = A’ ∪ B’
• (A ∪ B)’ = A’ ∩ B’

### Find the probability

1. Define the experiment
2. List the simple events associated with the experiment and test each to make certain that it cannot be decomposed; this defines the sample space S
3. Assign reasonable Probabilities to the outcomes in S
4. Define the event of interest, A, as a union of simple events or a collection of outcomes
5. Fine P(A) by summing the probabilities of the simple events whose outcomes are in A