Conditional Probability & Independence

The conditional probability formula

The formula is $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ — the probability both events happen, divided by the probability of the given event. The given event becomes the new denominator because the condition throws away every outcome where $B$ didn't happen.

You can often skip the formula and just restrict the sample space directly. Given that a card is a face card, the world shrinks to the $12$ face cards. Four of them are kings, so $P(\text{king} \mid \text{face card}) = \dfrac{4}{12} = \dfrac{1}{3}$. Count inside the condition, then divide by the size of the condition.

Testing for independence

Events $A$ and $B$ are independent exactly when $P(A \mid B) = P(A)$ — the condition told you nothing new. Equivalently, $A$ and $B$ are independent when $P(A \cap B) = P(A) \cdot P(B)$. Check either equation with the numbers you're given; if it holds, the events are independent.

Physical separation is the usual tell: separate coin flips, separate spins, a die roll and a card draw. Drawing twice without replacement is the classic dependent situation — the first draw changes what's left for the second. The tree below shows two independent coin flips branching into four equally likely outcomes.

Independent is not mutually exclusive

These two words get swapped constantly, and they mean nearly opposite things. Mutually exclusive events cannot both happen: $P(A \cap B) = 0$. Independent events can both happen — their probabilities just multiply: $P(A \cap B) = P(A) \cdot P(B)$, which is not $0$ unless one of the events was impossible to begin with.

In fact, two events with nonzero probabilities that are mutually exclusive are automatically dependent: if one happens, the other's probability drops to $0$, which is a change.

Worked examples

Try one yourself

Common questions