Problems where you are given the probabilities of multiple events, their unions and/or intersections, or conditional probabilities can typically be solved by combining the given terms using the axioms of probability or any of the definitional formulas of probabilities. First, express the story problem in terms of given event probabilities and their unions or intersections. Note you can often find complements as 1 minus the given probability. It may be helpful to draw a Venn diagram to organize the probabilities of each union. In some cases you can answer directly from this diagram. Use any of the definitional formulas for probabilities to set the term you are interested in equal to the terms you are given. Populate with given probabilities and calculate the necessary value. ## Joint probability $P(A \cap B) = P(A) + P(B) - P(A \cup B)$ ## Conditional Probability $P(A|B) = \frac{P(A \cap B)}{P(B)}$ ## Law of Total Probability $P(A) = P(A|B)P(B) + P(A|B')*P(B')$ ## Bayes Rule $P(A|B) = \frac{P(B|A) * P(A)}{P(B)}$ ## Union of multiple events $P(\text{Union of many events}) = P(\text{Singles}) - P(\text{Doubles}) + P(\text{Triples}) - P(\text{Quadruples}) \dots$