Dependent Events
Dependent events are events where the outcome of one affects the probability of the other.
Formula
P(A \text{ and } B) = P(A) \times P(B|A)
Definition
Dependent events are events where what happens first changes the probability of what happens next: one event affects the other. Events $A$ and $B$ are dependent if $P(A \text{ and } B) \neq P(A)P(B)$, meaning $P(B|A) \neq P(B)$, and the multiplication rule for dependent events is $P(A \text{ and } B) = P(A)P(B|A)$. Formally, dependence is equivalent to $\text{Cov}(\mathbb{1}_A, \mathbb{1}_B) \neq 0$, where $\mathbb{1}_A$ is the indicator function of $A$; conditional probability $P(B|A) = P(A \cap B)/P(A)$ captures the updated probability of $B$ given $A$, and dependence structures in multivariate distributions are captured by copulas.
Example
Drawing two cards without replacing the first: if you draw a king first, there are now only $3$ kings left in the $51$ remaining cards, so the first draw affected the probability of the second. A bag with $5$ red and $3$ blue marbles has $P(\text{red first}) = 5/8$; after removing a red marble, $P(\text{red second} \mid \text{red first}) = 4/7$, so $P(\text{both red}) = 5/8 \times 4/7 = 5/14$. Sampling without replacement from a finite population creates dependent events: for a population of $N$ items with $K$ successes, drawing $n$ times without replacement follows a hypergeometric distribution, $P(X=k) = \binom{K}{k}\binom{N-K}{n-k}/\binom{N}{n}$, which differs from the binomial (sampling with replacement/independence).
Key Insight
Drawing without replacement always creates dependent events, while drawing with replacement creates independent events; many real-world events are dependent, such as weather today and tomorrow, medical test results and disease presence, or stock returns on consecutive days, so recognizing dependence is crucial for correct probability calculations. Markov chains model a special form of dependence where the future depends only on the present state, not the full history, $P(X_{n+1} \mid X_n, X_{n-1}, \ldots) = P(X_{n+1} \mid X_n)$, enabling tractable analysis of many real-world dependent processes.