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This question is commonly asked for Software Engineer candidates at Optiver during technical interviews.

Design dice betting strategy under time limit

Quick Overview

This question evaluates probabilistic reasoning, expected-value computation, risk management and staking-strategy concepts including Kelly-style bet sizing, confidence-interval estimation, and return–variance trade-offs in the Statistics & Math domain.

Repeated Dice Betting with Time and Risk Constraints

You have T = 10 minutes and an initial bankroll B = $1,000 to play a repeated betting game with a fair six‑sided die. Each roll is independent and takes 3 seconds, so you can make at most N = floor(10×60 / 3) = 200 bets.

Before each roll, you may stake any fraction of your current bankroll on exactly one of the following wagers. Payoffs are net profits per $1 staked (you also lose your stake on a loss):

(A) “roll is 6” with probability p = 1/6, pays b = 6:1
(B) “roll is 5 or 6” with p = 2/6 = 1/3, pays b = 2.2:1
(C) “roll is odd” with p = 3/6 = 1/2, pays b = 1.1:1

Tasks:

Which wager(s) have positive expected value? Show EV calculations.
For your chosen wager, compute the optimal Kelly bet fraction that maximizes expected log wealth, and propose a more conservative fraction that keeps the probability of a 50% drawdown below 5%.
Outline a practical staking policy for this time‑limited setting and explain when you would stop betting.
After n = 60 “odd” bets, you observe k = 36 wins. Provide a fast 95% confidence interval for the win probability p and interpret it.
Discuss the return–variance trade‑offs, and how your allocation changes if rolls slow down or if T is shorter.

Quick Overview

Repeated Dice Betting with Time and Risk Constraints

Before each roll, you may stake any fraction of your current bankroll on exactly one of the following wagers. Payoffs are net profits per $1 staked (you also lose your stake on a loss):

(A) “roll is 6” with probability p = 1/6, pays b = 6:1

(B) “roll is 5 or 6” with p = 2/6 = 1/3, pays b = 2.2:1

Tasks:

Which wager(s) have positive expected value? Show EV calculations.

For your chosen wager, compute the optimal Kelly bet fraction that maximizes expected log wealth, and propose a more conservative fraction that keeps the probability of a 50% drawdown below 5%.

Outline a practical staking policy for this time‑limited setting and explain when you would stop betting.

After n = 60 “odd” bets, you observe k = 36 wins. Provide a fast 95% confidence interval for the win probability p and interpret it.

Discuss the return–variance trade‑offs, and how your allocation changes if rolls slow down or if T is shorter.

Design dice betting strategy under time limit

Quick Overview

Repeated Dice Betting with Time and Risk Constraints

Solution

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Design dice betting strategy under time limit

Quick Overview

Repeated Dice Betting with Time and Risk Constraints

Solution

Comments (0)