Independent events
Two events are independent if the occurrence of one does not affect the probability of the other. Tossing a coin and rolling a die are independent. For independent events the multiplication rule holds:
Key formula
$P(A \cap B) = P(A) \times P(B)$
Example
If $P(A) = \frac{1}{2}$ and $P(B) = \frac{1}{3}$ are independent, then $P(A \cap B) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$.
Dependent events
Two events are dependent if the outcome of the first changes the probability of the second. Drawing objects without replacement creates dependent events, because the second probability is based on the reduced set.
Example
A bag holds $5$ red and $3$ blue balls ($8$ total). Drawing two red balls without replacement: $P = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$. With replacement they would be independent: $\frac{5}{8} \times \frac{5}{8} = \frac{25}{64}$.
Choosing the right probability
- With replacement → independent → probabilities stay the same.
- Without replacement → dependent → reduce the numerator and denominator for the second draw.
Remember
“And” usually means multiply. Always ask whether the first event changes the second before deciding which probabilities to use.