Untitled Notes

Basic Concepts and Axiomatic Definition

Random experiment: a process with unpredictable outcomes
- Examples: tossing a coin, rolling dice, selecting cards
Sample space: collection of all possible outcomes
- Can be specified using roster method or rule method
Event: subset of the sample space with a defined probability
- Examples: observing odd/even numbers, less than 3 dots on a die

Probability provides a framework to analyze uncertainty in real-world events
Helps in decision making and statistical inference
Central to hypothesis testing and assessing whether observed values represent the hypothesis or occurred by chance

Probability of an event ranges from 0 to 1
Sum of probabilities of all possible outcomes is 1
Complement of an event, intersection, and union of events can be used to define probabilities

Conditional probability: probability of an event given that another event has occurred
Independence: events that do not affect each other's probabilities

In the 17th century, statistics was considered an "art" without a mathematical foundation
Probability was first explored in gambling and later integrated with inferential statistics
Statistics became recognized as a vital tool in various fields of research

Intersection of n events: 𝐴1 ∩ 𝐴2 ∩ ⋯ ∩ 𝐴𝑛
- Collection of sample points that belong in each one of A1, A2, ..., An
- Occurs if all of the n events occurred
Union of n events: 𝐴1 ∪ 𝐴2 ∪ ⋯ ∪ 𝐴𝑛
- Collection of sample points that belong in at least one of A1, A2, ..., An
- Occurs if at least one of the n events occurred

Event A: Same number of dots on both dice
- A = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
Event B: 3 dots on the red die
- B = { 1,3 , 2,3 , 3,3 , 4,3 , 5,3 , 6,3 }
Event C: Sum of dots on both dice is 5
- C = 1,4 , 2,3 , 3,2 , 4,1
Event D: 7 dots on the green die
- D = 𝑜𝑟 𝜙, the empty set
Other events:
- AC = {(1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5)}
- A ∪ B = {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1,3),(2,3),(4,3),(5,3),(6,3)}
- A ∩ B = {(3,3)}
- A ∪ B ∪ D = {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1,3),(2,3),(4,3),(5,3),(6,3)}
- A ∩ B ∩ D = ∅

Two events A and B are mutually exclusive if and only if A ∩ B = ∅
Mutually exclusive events can be extended to more than two events
Any collection of events is mutually exclusive if the collection is pairwise disjoint

Set theory:
- A complement, AC: Event A will not occur
- Universal Set, Ω: Sure event
Probability theory:
- Event A will occur
- Event A or B will occur (A union B, A ∪ B)
- At least one of A, B, and C will occur (A union B union C, A ∪ B ∪ C)
- Events A and B will occur (A intersection B, A ∩ B)
- All events A, B, and C will occur (A intersection B intersection C, 𝐴 ∩ 𝐵 ∩ 𝐶)
- Only event A will occur but not event B (A intersection B complement, A ∩ BC)
- Events A and B are mutually exclusive

Probability of an event A, denoted by 𝑃(𝐴), assigns a measure of chance that event A will occur
Properties:
- 𝑃 𝐴 ≥ 0 for any event 𝐴
- 𝑃 𝛺 = 1
- Finite Additivity: If A = A1 ∪ A2 ∪ ⋯ ∪ An and A1, A2, ..., An are mutually exclusive, then 𝑃(𝐴) = 𝑃(𝐴1) + 𝑃(𝐴2) + ⋯ + 𝑃(𝐴𝑛)

Example of using set notation to represent probabilities
- 𝛺 = 𝐴 ∪ 𝐵 ∪ 𝐶
- 𝑃(𝛺) = 𝑃(𝐴 ∪ 𝐵 ∪ 𝐶)
Calculation of probabilities using the definition of probability
- LHS: 𝑃 𝛺 = 1
- RHS: 𝑃 𝐴⋃𝐵⋃𝐶 = 𝑃 𝐴 + 𝑃 𝐵 + 𝑃(𝐶)
Conclusion: 𝑃 𝐴 + 𝑃 𝐵 + 𝑃 𝐶 = 1
Use of algebra to solve for the corresponding probabilities
Goal: Compute 𝑃(𝐴 ∪ 𝐵)

Example of calculating probabilities using the definition of probability or Finite Additivity
Given probabilities:
- 𝑃 𝐴 = 1/6
- 𝑃 𝐵 = 𝑃 𝐴 = 1/6
- 𝑃 𝐶 = 4𝑃 𝐴 = 4/16 = 4/6
Calculation of 𝑃(𝐴 ∪ 𝐵) using the definition of probability or Finite Additivity
- 𝑃 𝐴⋃𝐵 = 𝑃 𝐴 + 𝑃 𝐵 = 1/6 + 1/6 = 2/6 ≈ 0.33