Formula: P(A) = Number of favorable outcomes / Total number of possible outcomes
Scenario: A jar contains 5 red, 3 blue, and 2 green marbles. What's the probability of drawing two red marbles in succession without replacement?
Explanation: First draw has 5 red out of 10 total marbles. Second draw has 4 red out of 9 remaining marbles. Multiply these probabilities.
Formula: P(A|B) = P(A∩B) / P(B)
Scenario: A weather forecast predicts rain with 80% accuracy when it actually rains, and 90% accuracy when it doesn't. In this region, it rains 25% of days. What's the probability it will actually rain when the forecast predicts rain?
Explanation: - P(A∩B): Probability it rains AND forecast predicts rain (0.25 * 0.80 = 0.20) - P(B): Total probability forecast predicts rain (true positives + false positives = 0.20 + 0.675 = 0.275)
Formula: C(n, k) = n! / (k!(n-k)!)
Scenario: How many ways can you form a 3-person committee from 10 people?
Explanation: Order doesn't matter in committee selection - the same people in different orders count as the same committee.
Formula: P(n, k) = n! / (n-k)!
Scenario: In a race with 8 runners, how many possible gold-silver-bronze podium outcomes are there?
Explanation: Order matters - who gets gold is different from who gets silver or bronze.