Compute expected payoff of reroll dice game

Q: Compute expected payoff of reroll dice game

This question evaluates understanding of probability theory and expected-value computation in stopping-time processes, including conditioning and infinite-series reasoning, and also tests modeling of per-trial costs and break-even analysis.

Q: How do I approach Statistics & Math interview questions?

Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

Q: What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during Online Assessment rounds at Citibank.

Q: What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Citibank during technical interviews.

Question

Repeated Die Rolls with Stopping Rule and Costs

You repeatedly roll a fair six-sided die. After each roll you immediately receive a payout equal to the face value. If you roll 4, 5, or 6 you must roll again; the game stops the first time you roll 1, 2, or 3.

Tasks:

Fair die (uniform): Derive the expected total payout. Show all steps (either by conditioning on the stopping rule or by using an infinite series).
General biased die: Let p = P(roll in {4,5,6}) and let μ be the expected value of a single roll. Express the expected total payout in terms of p and μ.
Per-roll cost: Suppose each roll after the first incurs a fixed cost c. Derive the break-even c that makes the game fair (zero expected net).

Assumptions:

Rolls are independent and identically distributed.
Payout is collected every time a roll occurs; the process stops at the first roll in {1,2,3}.

Compute expected payoff of reroll dice game

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Compute expected payoff of reroll dice game

Repeated Die Rolls with Stopping Rule and Costs

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