Probability

Probability is crucial in environments like the stock market and casino games, as it provides a systematic way to quantify and consider uncertainty arising from unknown variables and "noise."

It quantifies uncertainty from underlying random processes, which are scenarios with known possible outcomes but unknown specific results (e.g., coin tosses, dice rolls).
Probability is a numerical measure (0 \leq P(A) \leq 1) of the likelihood of an outcome.
The Frequentist Interpretation defines probability as the proportion of occurrences over an infinite number of trials.

Law of Large Numbers: As more observations occur, the observed frequency of an outcome approaches its expected probability (e.g., P(2) \rightarrow \frac{1}{6} for a die roll).
Gambler’s Fallacy: The misconception that past independent events influence future probabilities (e.g., after 10 heads, the P(H) for the next flip remains 0.5).

Sample Space (Ω): All possible outcomes of an experiment (e.g., {1, 2, 3, 4, 5, 6} for a die).
Event (A): A subset of the sample space (e.g., even sides A = {2, 4, 6}).
Null Event (∅): An empty event where no outcomes occur.

Union (∪): Combines elements from events A and B (e.g., A ∪ B).
Intersection (∩): Represents the common elements between events A and B (e.g., A ∩ B).
Disjoint Outcomes: Events that cannot occur simultaneously (e.g., a coin cannot be both heads and tails). For disjoint events, P(A \text{ or } B) = P(A) + P(B).
Non-disjoint Outcomes: Events that can occur simultaneously.
Complement (A^c): The event that A does not occur. P(A) + P(A^c) = 1.
General Addition Rule: For any two events A and B, P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B).

Lists all potential events and their probabilities, adhering to three rules:
1. Events must be disjoint.
2. Each probability must be between 0 and 1.
3. The total probability must equal 1.

Independent Processes: One event's outcome doesn't inform another's (e.g., consecutive coin tosses).
Dependent Processes: One event's outcome affects another's probability (e.g., drawing cards without replacement).
Events A and B are independent if P(A | B) = P(A).

Marginal Probability: The likelihood of a standalone event.
Joint Probability: The probability of two events occurring concurrently (e.g., P(\text{relapsed and desipramine}) \approx 0.14).
Conditional Probability: The probability of one event given another has occurred. Expressed as P(A | B) = \frac{P(A \text{ and } B)}{P(B)} (e.g., P(\text{relapse | desipramine}) \approx 0.42).

Mutually Exclusive: Cannot happen at the same time (e.g., being male and female).
Independent: Do not affect each other's likelihood (e.g., two unrelated coin flips).