Fundamentals of Mathematical Probability Theory
Probability is the quantitative measurement of the likelihood that an event will occur in a random experiment, bounded on the real interval $[0, 1]$, where 0 signifies an impossible event and 1 signifies absolute certainty.
Core Probability Theorems and Rules
| Probability Concept | Standard Formula | Description |
|---|---|---|
| Complement Rule | $P(A') = 1 - P(A)$ | Probability that event A does not happen |
| Multiplication Rule (Independent) | $P(A \cap B) = P(A) \times P(B)$ | Probability that both events A and B happen together |
| Addition Rule (General) | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ | Probability that at least one event (A or B) occurs |
| Binomial Probability Mass Function | $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ | Probability of exactly $k$ successes in $n$ Bernoulli trials |