Probability — common mistakes

A common exam question asks you to determine if two events are independent. To do this, calculate $P(A)$, $P(B)$, and $P(A \cap B)$ from the information given. Then, check if $P(A \cap B)$ is equal to $P(A) \times P(B)$. If they are equal, the events are independent. If not, they are dependent. Do not just assume independence unless the question explicitly states it or the context (e.g., 'with replacement') makes it obvious.

Mutually exclusive events cannot happen at the same time ($P(A \cap B) = 0$). Independent events can happen at the same time, but the occurrence of one does not change the probability of the other ($P(A \cap B) = P(A) \times P(B)$). Think of rolling a die: 'getting a 1' and 'getting a 6' are mutually exclusive. Tossing a coin twice: 'getting a head on the first toss' and 'getting a head on the second toss' are independent.

Venn diagrams are excellent for visualising the relationships (union, intersection, complement) between two or three events in a single experiment, especially for problems involving counts or the addition rule. Tree diagrams are best for multi-stage experiments where outcomes are sequential and often conditional, like drawing items without replacement. They help you systematically list all possible outcomes and calculate their probabilities by multiplying along the branches.

You ADD probabilities when you are considering alternative outcomes (Event A OR Event B). This corresponds to the union of events, $P(A \cup B)$. You MULTIPLY probabilities when you are considering a sequence of events (Event A AND THEN Event B). This corresponds to the intersection of events, $P(A \cap B)$. Be careful to use the correct multiplication rule (for independent vs. dependent events).