If any of these conditions are violated, the given data set is inconsistent, implying an error in data collection or reporting.

Experiment: Any process that produces an observation or outcome.
Outcome: A single result of an experiment.
Sample Space ($S$ or $\Omega$): The set of all possible outcomes of a random experiment.
Event ($E$): A subset of the sample space; a collection of outcomes.
- Simple Event: An event consisting of a single outcome.
- Compound Event: An event consisting of more than one outcome.
Probability of an Event ($P(E)$): A numerical measure of the likelihood that an event will occur.
- Classical Definition: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ (assuming equally likely outcomes).
- Relative Frequency Definition: $P(E) = \frac{\text{Number of times E occurs}}{\text{Total number of trials}}$ (as number of trials approaches infinity).
- Subjective Definition: Based on personal judgment.
Axioms of Probability:
- $0 \le P(E) \le 1$ for any event $E$.
- $P(S) = 1$ (The probability of the sample space occurring is 1).
- If $E_1, E_2, ...$ are mutually exclusive events, then $P(E_1 \cup E_2 \cup ...) = P(E_1) + P(E_2) + ...$.
Complement of an Event ($E^c$ or $\bar{E}$): The event that $E$ does not occur. $P(E^c) = 1 - P(E)$.
Mutually Exclusive Events: Events that cannot occur at the same time ($A \cap B = \emptyset$).
Independent Events: Events where the occurrence of one does not affect the probability of the other.

For any two events A and B (not necessarily mutually exclusive): $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
- $P(A \cup B)$: Probability that A occurs OR B occurs OR both occur.
- $P(A \cap B)$: Probability that A and B both occur (intersection).
If A and B are mutually exclusive events: $$P(A \cup B) = P(A) + P(B)$$
For three events A, B, C: $$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$$

For any two events A and B: $$P(A \cap B) = P(A) P(B|A)$$ or $$P(A \cap B) = P(B) P(A|B)$$
- This is used to find the probability that both A and B occur.
If A and B are independent events: $$P(A \cap B) = P(A) P(B)$$
- The occurrence of one event does not change the probability of the other.

The probability of an event A occurring given that another event B has already occurred ($P(A|B)$). It effectively reduces the sample space to event B.
Formula: $$P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{provided } P(B) > 0$$ Similarly, $$P(B|A) = \frac{P(A \cap B)}{P(A)}, \quad \text{provided } P(A) > 0$$
If A and B are independent, then $P(A|B) = P(A)$ and $P(B|A) = P(B)$.

A fundamental theorem in probability that describes how to update the probability of a hypothesis ($A_i$) based on new evidence ($B$). It relates the conditional probability $P(A|B)$ to $P(B|A)$.
Let $A_1, A