Compute conditional probabilities and determine whether events are independent.
The conditional probability P(A | B) is the probability of event A given that event B has already occurred. It narrows the sample space to only those outcomes where B occurs.
Independent Events
Dependent Events
Always test independence by checking if P(A∩B) = P(A)·P(B).
Deck of 52. P(King on 2nd draw | King on 1st draw) = 3/51 = 1/17, since only 3 kings remain from 51 cards. Compare to P(King) = 4/52 = 1/13 — different, so the events are dependent.
Remember This!
For independent events, "with replacement" sampling ensures each draw is independent. "Without replacement" creates dependent events because the sample space changes.
P(A) = 0.4, P(B) = 0.5, P(A∩B) = 0.2. Are A and B independent?