Waiting for the First Heart: Exploring Geometric Distributions
Task
Lesson Description In this lesson, students use the Tuva Simulation environment to model a waiting-time situation using a geometric distribution. By simulating repeated card draws with replacement, students investigate how long it typically takes for the first success to occur. Students interpret simulated results to estimate probabilities of events such as the game ending on a specific draw, within a given number of draws, or lasting beyond a threshold, without relying on formal probability formulas. Learning Objectives Students will be able to: Identify and define a geometric random variable in a real-world context Interpret the probability of success in a repeated-trial situation Use simulation to estimate probabilities involving a geometric distribution Describe how the shape of a geometric distribution reflects waiting-time behavior Interpret probability statements involving “exactly,” “at most,” and “at least” in context Explain how increasing the number of simulations affects the stability of results
Introduction A game involves repeatedly drawing one card from a standard 52-card deck with replacement. The game continues until the player draws a heart for the first time. At that point, the game stops. The game is designed so that: The player earns a bonus if the game ends by the 5th draw, and The player loses interest if the game lasts beyond 5 draws. The designer of the game wants to understand how long players typically have to wait for the first heart. You are interested in the following questions: 1. What is the probability that the game ends exactly on the 5th draw? 2. What is the probability that the game ends within the first 5 draws? 3. What is the probability that the game lasts 5 draws or longer?
Topics and standards: Math, Simulation, Probability & Random Variables, Relative Frequency, Statistics & AP Statistics, Assessment, AP Statistics