# Solve Three Probability Models Solve three independent probability questions. Show the sample space or stochastic assumptions in each part and give an exact result. ### Constraints & Assumptions - Every die roll is independent and uniformly distributed from 1 through 6. - In the meeting problem, each person arrives independently and uniformly during a 60-minute interval and waits at most 15 minutes. - In the traffic problem, vehicle arrivals follow a homogeneous Poisson process with independent increments. ### Clarifying Questions to Ask - In the die game, is the payoff the face on the final accepted roll rather than the sum? - Does a meeting occur when one person arrives exactly 15 minutes after the other? - Is the one-hour traffic probability assumed stationary across both half-hours? ### Part 1 — Three-Roll Optimal Stopping You may roll a fair die at most three times. After each roll you may stop and receive that face value; if you continue, the previous face is discarded. Compute the optimal policy and expected payoff. #### What This Part Should Cover - Backward induction and the acceptance threshold on each remaining decision. - An exact expected value, including tie behavior. ### Part 2 — Two People Meeting Two people arrive independently during one hour and each waits 15 minutes. Compute the probability that they meet. #### What This Part Should Cover - A geometric sample space and the complement formed by two corner triangles. - Correct scaling of 15 minutes to one quarter of the hour. ### Part 3 — Poisson Half-Hour Probability The probability of at least one vehicle in an hour is 0.96. Compute the probability of at least one vehicle in half an hour. #### What This Part Should Cover - Use of the no-arrival complement and independent equal-length increments. - A result derived without first needing the explicit rate parameter. ```hint Work backward on the die The value of rejecting a roll is the expected value of the game with one fewer roll remaining. ``` ```hint Draw the arrival square Meeting corresponds to points within distance one quarter of the diagonal line. ``` ### What a Strong Answer Covers - Exact probability reasoning, clearly stated independence assumptions, and checks that every result lies in its feasible range. ### Follow-up Questions - How does the die policy change with N allowed rolls? - What is the meeting probability when the two waiting times differ?