Three Probability Models to Solve

Work through three separate probability questions. For each part, state the sample space or stochastic assumptions and provide an exact answer.

Constraints and Assumptions

• Each die roll is independent and uniform over the values 1 to 6.
• For the meeting problem, both people arrive independently and uniformly within a 60-minute window, and each waits no more than 15 minutes.
• For the traffic problem, vehicle arrivals come from a homogeneous Poisson process with independent increments.

Questions to Clarify

• In the die game, is the payoff the value shown on the last accepted roll, not the total of all rolls?
• If one person arrives exactly 15 minutes after the other, does that count as a meeting?
• Is the one-hour traffic probability assumed to be stationary over the two half-hour periods?

Part 1 — Optimal Stopping with Three Rolls

You can roll a fair die up to three times. After any roll, you may stop and collect the face value shown; if you keep going, the prior roll is thrown away. Find the optimal stopping rule and the expected payoff.

This Part Should Cover

• Backward induction and the acceptance cutoff for each remaining decision.
• An exact expected value, including how ties are handled.

Part 2 — Meeting Probability for Two People

Two people arrive independently within a one-hour period, and each person waits for 15 minutes. Compute the probability that their waiting intervals overlap.

This Part Should Cover

• A geometric sample space, with the complement made up of two corner triangles.
• Correct conversion of 15 minutes to one quarter of the hour.

Part 3 — Poisson Probability for a Half Hour

The chance of at least one vehicle arriving during an hour is 0.96. Compute the chance of at least one vehicle arriving during a half hour.

This Part Should Cover

• Use of the no-arrival complement and independent increments over equal-length intervals.
• A result obtained without first solving for the explicit rate parameter.

Hint — Work backward for the die: The value of turning down a roll equals the expected value of the game with one fewer roll left.

Hint — Sketch the arrival square: A meeting corresponds to points whose distance from the diagonal line is at most one quarter.

What a Strong Answer Should Cover

• Exact probability reasoning, explicit independence assumptions, and verification that each result falls within its valid range.

Follow-Up Questions

• How does the die policy change if $N$ rolls are allowed?
• What is the meeting probability if the two waiting times are different?

Overview: Work through three independent probability questions. State the assumptions and derivation clearly, check edge cases, and show how each result changes when those assumptions are relaxed.