Week 8 - Probability I - Axioms and kinds

Question 1

Probability measure

Answer 1

A function that assigns a probability to each proposition, denoted by Pr(…). It should adhere to the Kolmogorov Axioms to be considered a valid probability measure.

Question 2

Kolmogorov Axioms

Answer 2

Axiom 1: Pr(p) is a number between 0 and 1.
Axiom 2: If p is certain, Pr(p) = 1.
Axiom 3: If p and q are incompatible, Pr(p or q) = Pr(p) + Pr(q). These axioms ensure that the probability measure is well-defined and behaves consistently.

Question 3

Counting outcomes

Answer 3

In cases with a finite number of equally probable outcomes, the probability of a proposition can be calculated by counting the number of outcomes where the proposition occurs and dividing by the total number of possible outcomes. This method is useful for simple cases, such as rolling dice, flipping coins, or drawing cards from a deck.

Question 4

Subjective probability

Answer 4

Also known as credences, degrees of belief, or personal probabilities, subjective probabilities represent an individual’s degree of belief in a particular outcome. They may vary from person to person based on their knowledge or perspective.

Question 5

Objective probability

Answer 5

Objective probabilities, also known as chances or statistical chances, represent real or objective tendencies for certain outcomes to occur. Examples include the half-life of radium or the probability of developing pancreatic cancer. These probabilities are independent of individual beliefs or perspectives.

Question 6

Certain/incompatible propositions

Answer 6

Certain propositions have a probability of 1, meaning they are guaranteed to occur. Incompatible propositions cannot both be true at the same time. If p and q are incompatible, Pr(p and q) = 0, and Pr(p or q) = Pr(p) + Pr(q).

Question 7

Six properties of probability

Answer 7

1. Pr(p) + Pr(not-p) = 1.
2. If p and q are logically equivalent, Pr(p) = Pr(q).
3. Pr(p) = Pr(p and q) + Pr(p and not-q).
4. Pr(p) ≥ Pr(p and q).
5. Pr(p) ≤ Pr(p or q).
6. Pr(p or q) = Pr(p) + Pr(q) - Pr(p and q).

These properties help derive relationships between probabilities of different propositions and ensure the consistency of the probability measure.