Geometric: Waiting Time

Probability · Axiom Academy

EXAMPLE Geometric Distribution: Waiting Time A basketball player makes free throws with probability 0.8. Find the probability the first miss occurs on the 5th shot, and the expected number of shots until a miss. Excellent work! You've completed this geometric distribution example. Here's what we learned: Geometric distribution: Models the number of trials until the first success (or failure). Here we waited for the first miss (failure). Memoryless property: The probability formula is P(X = k) = (1-p)^(k-1) × p, where p is the probability of "success" (the event we're waiting for). Expected value: For a geometric distribution, E(X) = 1/p. With p = 0.2 (miss probability), we expect the first miss after 5 shots on average. Key insight: To get the first miss on the 5th shot, we need 4 consecutive makes (0.8^4) followed by 1 miss (0.2). This distribution is perfect for "waiting time" problems where you count trials until an event occurs. Practice with different scenarios!

This is the written version of the interactive lesson above. See the full Probability course.