Full notes on probability

Probability is the mathematical measure of the likelihood of an event occurring. The sample space ($\Omega$) is the set of all possible outcomes of an experiment, and an event ($A$) is a subset of these outcomes. The probability of an event $A$ is given by:

$$P(A) = \frac{\text{Number of outcomes in } A}{\text{Total number of outcomes in } \Omega}$$

Probabilities always range from 0 (impossible event) to 1 (certain event), i.e., $0 \le P(A) \le 1$. The complement of an event $A$, denoted $A'$, is the event that $A$ does not occur, and $P(A') = 1 - P(A)$.

The Addition Rule is used for the probability of event $A$ or event $B$ occurring, denoted $P(A \cup B)$. If $A$ and $B$ are mutually exclusive (cannot happen at the same time), then $P(A \cup B) = P(A) + P(B)$. Otherwise, for any two events:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

where $P(A \cap B)$ is the probability of both $A$ and $B$ occurring.

The Multiplication Rule is for the probability of event $A$ and event $B$ occurring. For independent events (where the occurrence of one does not affect the other), $P(A \cap B) = P(A) \times P(B)$. For dependent events, we use conditional probability, $P(A|B)$, which is the probability of $A$ occurring given that $B$ has already occurred:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

From this, the general multiplication rule is $P(A \cap B) = P(A|B) \times P(B)$.