Compute expected payoff of reroll dice game
Repeated Die Rolls with Stopping Rule and Costs
You are presented with a game involving repeated rolls of a fair six-sided die. After each roll, you immediately receive a payout equal to the face value of the roll. The game has a specific stopping rule: if you roll a 4, 5, or 6, you must roll again. The game concludes the first time you roll a 1, 2, or 3.
Tasks
- Fair Die (Uniform Probability): Derive the expected total payout for this game. Your derivation should clearly show all steps, either by conditioning on the stopping rule or by using an infinite series approach.
- General Biased Die: Generalize the problem. Let
prepresent the probability of rolling a 4, 5, or 6 (i.e.,p = P(roll ∈ {4, 5, 6})), and letμbe the expected value of a single roll. Express the expected total payout in terms ofpandμ. - Per-Roll Cost: Introduce a fixed cost
cfor each roll after the first roll. Derive the break-even value ofcthat makes the game fair (i.e., results in a zero expected net payout).
Assumptions
- All die rolls are independent and identically distributed (IID).
- The payout corresponding to the face value is collected immediately after every roll.
- The game process stops precisely at the first roll that results in a 1, 2, or 3.
AI interview prep
Core dilemma
Accurately calculating expected values in a sequential, probabilistic process with a conditional stopping rule, varying costs, and generalizing the solution for biased probabilities.
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Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Compute expected payoff of reroll dice game
Repeated Die Rolls with Stopping Rule and Costs
You are presented with a game involving repeated rolls of a fair six-sided die. After each roll, you immediately receive a payout equal to the face value of the roll. The game has a specific stopping rule: if you roll a 4, 5, or 6, you must roll again. The game concludes the first time you roll a 1, 2, or 3.
Tasks
- Fair Die (Uniform Probability): Derive the expected total payout for this game. Your derivation should clearly show all steps, either by conditioning on the stopping rule or by using an infinite series approach.
- General Biased Die: Generalize the problem. Let
prepresent the probability of rolling a 4, 5, or 6 (i.e.,p = P(roll ∈ {4, 5, 6})), and letμbe the expected value of a single roll. Express the expected total payout in terms ofpandμ. - Per-Roll Cost: Introduce a fixed cost
cfor each roll after the first roll. Derive the break-even value ofcthat makes the game fair (i.e., results in a zero expected net payout).
Assumptions
- All die rolls are independent and identically distributed (IID).
- The payout corresponding to the face value is collected immediately after every roll.
- The game process stops precisely at the first roll that results in a 1, 2, or 3.