PROBABILITY

 Certainly! Here are some basic properties of probability that are fundamental to understanding how probabilities work:

1. **Probability Range**: The probability of any event happening lies between 0 and 1, inclusive. That is, 0 ≤ P(event) ≤ 1.

2. **Certain Event and Impossible Event**: The probability of a certain event is 1, while the probability of an impossible event is 0. For example, the probability of the sun rising tomorrow is very close to 1, and the probability of rolling a 7 on a standard six-sided die is 0.

3. **Complementary Probability**: The probability of the complement of an event (i.e., the event not happening) is 1 minus the probability of the event itself. This can be written as P(not A) = 1 - P(A).

4. **Addition Rule for Disjoint (Mutually Exclusive) Events**: If two events A and B are disjoint (they cannot happen simultaneously), then the probability of either A or B happening is the sum of their individual probabilities: P (A or B) = P(A) + P(B).

5. **General Addition Rule**: For any two events A and B (not necessarily disjoint), the probability of either A or B happening is the sum of their individual probabilities minus the probability of their intersection: P (A or B) = P(A) + P(B) - P (A and B).

6. **Multiplication Rule for Independent Events**: If two events A and B are independent (the occurrence of one doesn't affect the occurrence of the other), then the probability of both A and B happening is the product of their individual probabilities: P (A and B) = P(A) * P(B).

7. **Conditional Probability**: The conditional probability of event A happening given that event B has occurred is denoted as P(A|B) and is calculated as the probability of both A and B happening divided by the probability of B: P(A|B) = P (A and B) / P(B).

8. **Multiplication Rule for Dependent Events**: For two events A and B that are dependent, the probability of both A and B happening is calculated using the conditional probability: P (A and B) = P(A|B) * P(B).

9. **Law of Total Probability**: If you have a set of mutually exclusive and exhaustive events {B1, B2, ..., Bn}, then the probability of event A happening can be calculated by summing up the probabilities of A happening given each of the events Bi, weighted by the probabilities of Bi occurring: P(A) = Σ[P(A|Bi) * P(Bi)] for i = 1 to n.

10. **Bayes' Theorem**: Bayes' theorem allows you to update the probability of an event based on new information. It's expressed as: P(B|A) = [P(A|B) * P(B)] / P(A), where B is a hypothesis and A is evidence.

These are some of the basic properties and rules of probability. They form the foundation for more complex calculations and applications of probability theory.