Dependent events

Property The outcome of the first event does affect the second event. The probability is $P(A \text{ and } B) = P(A) \cdot P(B)$, where $P(B)$ is calculated under the new conditions.

Examples From a bag with 5 red and 4 blue marbles, drawing two red without replacement: $P(\text{red, then red}) = \frac{5}{9} \cdot \frac{4}{8} = \frac{20}{72} = \frac{5}{18}$. Drawing two face cards from a deck without replacement: $P(\text{Face Card, then Face Card}) = \frac{12}{52} \cdot \frac{11}{51} = \frac{132}{2652} = \frac{11}{221}$. Choosing two students from 10 girls and 8 boys: $P(\text{girl, then boy}) = \frac{10}{18} \cdot \frac{8}{17} = \frac{80}{306} = \frac{40}{153}$.

Explanation Imagine grabbing a doughnut from a box and eating it. Now there's one less! The next person's choice is different because of what you did. The events are linked, so the probability changes for the second event based on the first outcome. It's a chain reaction of chances!