Calculate P(A ∩ B), P(A ∪ B), and conditional probabilities for independent and dependent events.
Joint probability is the probability that two or more events occur simultaneously. Denoted P(A ∩ B), it answers: "What's the chance both A and B happen?"
For independent events: P(A ∩ B) = P(A) × P(B)
Key Concepts:
Real-world examples:
Identify that the events are independent (one doesn't affect the other).
Multiply: P(A ∩ B) = P(A) × P(B)
Calculate P(A ∩ B) = P(A) × P(B|A), where P(B|A) depends on A occurring.
Use Bayes' theorem if needed: P(B|A) = P(A ∩ B) / P(A)
Use: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Subtract the intersection to avoid counting it twice.
P(First die = 3) = 1/6 ≈ 0.167 P(Second die = 5) = 1/6 ≈ 0.167 P(First=3 AND Second=5) = 1/6 × 1/6 = 1/36 ≈ 0.0278 Both dice rolls are independent—the first doesn't affect the second.
P(1st card is Ace) = 4/52 ≈ 0.077 P(2nd card is Ace | 1st was Ace) = 3/51 ≈ 0.059 P(Both Aces) = (4/52) × (3/51) ≈ 0.00452 The second draw depends on the first—fewer cards and aces remain.
P(A) = 0.3, P(B) = 0.5 (independent) P(A ∩ B) = 0.3 × 0.5 = 0.15 P(A ∪ B) = 0.3 + 0.5 − 0.15 = 0.65 There's a 65% chance at least one event occurs.
Related Tools
Basic probability calculations.
P(A|B) calculations.
Three event probabilities.
Conditional probability.
Probability-weighted average.
Decision theory.