Probability Formulae

• The Addition Rule is the fundamental principle for calculating the union of two events. It states that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, where the subtraction of the intersection is necessary to prevent double-counting the outcomes that belong to both sets.

• The Multiplication Rule governs the intersection of events and is derived from conditional logic. It is expressed as $P(A \cap B) = P(A) \times P(B|A)$, meaning the probability of both occurring is the probability of the first multiplied by the probability of the second given that the first has already happened.

• Conditional Probability ($P(A|B)$) measures the likelihood of an event occurring under the restriction that another event has already occurred. This effectively reduces the sample space from the entire universal set to only the outcomes contained within the 'given' event.

Applying the Formulae

• Step 1: Identify the Goal: Determine if the problem asks for an 'OR' (Union), 'AND' (Intersection), or 'GIVEN' (Conditional) probability to select the correct formula.

• Step 2: Check for Constraints: Verify if the events are Independent or Mutually Exclusive, as these conditions significantly simplify the standard formulae.

• Step 3: Algebraic Rearrangement: Often, you may need to solve for the intersection first using the Addition Rule before you can calculate a conditional probability. Rearranging $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ to find $P(A \cap B)$ is a common first step.

• Step 4: Visualization: Use Two-Way Tables for categorical data or Tree Diagrams for sequential events to map out probabilities before plugging them into the formulae.

• Intersection vs. Conditional: $P(A \cap B)$ is the probability of both events happening out of the entire sample space, whereas $P(A|B)$ is the probability of A happening out of only the outcomes where B is true.

• The 'Given That' Trigger: Whenever you see the phrase 'given that', immediately write down the conditional probability formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$. The term following 'given that' always goes in the denominator.

• Sanity Checks: Always ensure that any calculated probability is between $0$ and $1$. If you calculate a union $P(A \cup B)$ and it exceeds $1$, you likely forgot to subtract the intersection.

• Tree Diagram Logic: Remember that the probabilities on the second set of branches in a tree diagram are already conditional probabilities ($P(B|A)$). To find the intersection ($P(A \cap B)$), you must multiply along the branches.

• Venn Diagram Totals: When filling a Venn diagram, start from the center (the intersection) and work outwards. Subtract the intersection from the individual event totals to find the 'only A' and 'only B' regions.