Conditional Probability and Independence

Discrete Math · Axiom Academy

LESSON Conditional Probability and Independence Understanding how probabilities change with additional information, and when events don't influence each other Conditional probability measures the probability of event A occurring, given that event B has already occurred. We write this as P(A|B), read as "probability of A given B." What is P(A|B)? That is, if we know the card is a Spade, what's the probability it's an Ace? Solution: Once we know the card is a Spade, we're only considering 13 cards (the Spades). Of these 13, exactly 1 is an Ace. Therefore: Using the formula: P(A∩B) = 1/52 (Ace of Spades), P(B) = 13/52, so P(A|B) = (1/52)/(13/52) = 1/13 ✓ Two events A and B are independent if knowing that one occurred doesn't change the probability of the other. Formally: 4. Dependent vs. Independent Events Understanding the difference between dependent and independent events is crucial for calculating probabilities correctly. P(Ace|Spade) = 1/13 (as we calculated) Since P(Ace|Spade) = P(Ace), they are independent ! ✓

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