Dependent Events
Statistics & ProbabilityDependent 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.
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. The first draw affected the probability of the second.
Key Insight
Drawing without replacement always creates dependent events. Drawing with replacement creates independent events.
Definition
Events $A$ and $B$ are dependent if $P(A \text{ and } B) \neq P(A)P(B)$, meaning the occurrence of one affects the probability of the other: $P(B|A) \neq P(B)$. The multiplication rule for dependent events is $P(A \text{ and } B) = P(A)P(B|A)$.
Example
A bag has $5$ red and $3$ blue marbles. $P(\text{red first}) = 5/8$. After removing a red marble, $P(\text{red second} \mid \text{red first}) = 4/7$. $P(\text{both red}) = 5/8 \times 4/7 = 20/56 = 5/14$.
Key Insight
Many real-world events are dependent: weather today and weather tomorrow, medical test results and disease presence, stock returns on consecutive days. Recognizing dependence is crucial for correct probability calculations.
Definition
Events $A$ and $B$ are dependent if $P(A \cap B) \neq P(A)P(B)$, equivalently if $\text{Cov}(\mathbb{1}_A, \mathbb{1}_B) \neq 0$, 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 $A$. Dependence structures in multivariate distributions are captured by copulas.
Example
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
Markov chains model a special form of dependence where the future depends only on the present state, not the full history: $P(X_{n+1} \mid X_n, X_{n-1}, \ldots) = P(X_{n+1} \mid X_n)$. This Markov property enables tractable analysis of many real-world dependent processes.