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:

1. Which wager(s) have positive expected value? Show EV calculations.
2. 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%.
3. Outline a practical staking policy for this time‑limited setting and explain when you would stop betting.
4. After n = 60 “odd” bets, you observe k = 36 wins. Provide a fast 95% confidence interval for the win probability p and interpret it.
5. Discuss the return–variance trade‑offs, and how your allocation changes if rolls slow down or if T is shorter.