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.