Probability Formulae

Probability Range: The probability of any event $A$, denoted $P(A)$, must always fall between $0$ and $1$, where $0$ represents impossibility and $1$ represents certainty.

Sample Space: The sum of probabilities for all possible mutually exclusive and exhaustive outcomes in a sample space must equal $1$.

The Complement Rule: The probability of an event NOT occurring is denoted as $P(A')$ (or $P(A^c)$). It is calculated as $P(A') = 1 - P(A)$. This is particularly useful when calculating 'at least one' scenarios by finding the probability of 'none' and subtracting it from $1$.

General Addition Formula: For any two events $A$ and $B$, the probability that at least one occurs is given by $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

The Intersection: The term $P(A \cap B)$ represents the probability that both events occur simultaneously. We subtract it in the formula because it is counted twice—once within $P(A)$ and once within $P(B)$.

Mutually Exclusive Events: If two events cannot happen at the same time, they are mutually exclusive. In this case, $P(A \cap B) = 0$, and the formula simplifies to $P(A \cup B) = P(A) + P(B)$.

General Multiplication Formula: The probability of both $A$ and $B$ occurring is derived from the conditional probability formula: $P(A \cap B) = P(B) \times P(A|B)$.

Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. Mathematically, this means $P(A|B) = P(A)$.

Simplified Rule for Independence: For independent events, the multiplication rule simplifies to $P(A \cap B) = P(A) \times P(B)$. This is a common test for independence in exams.

Sanity Checks: Always ensure your final probability is between $0$ and $1$. If you get a negative number or a value greater than $1$, you likely forgot to subtract an intersection or used the wrong formula.

Visual Aids: Use Venn Diagrams for problems involving overlapping sets and Tree Diagrams for multi-stage experiments where outcomes depend on previous results.

Word Clues: Look for keywords like 'given that' (conditional), 'at least' (complement rule), and 'both' (intersection).

Verification: To prove independence, calculate $P(A) \times P(B)$ and check if it equals the given $P(A \cap B)$.