Betting Games: EV, Variance, Arbitrage, and Sizing

Context

You will be shown payout tables or odds for three independent games. Your task is to quickly evaluate value, risk, and trading opportunities. If exact odds are not provided, answer generically (in terms of probabilities p and decimal odds o) and illustrate with a small numeric example of your choice.

• Odds convention: use decimal odds o (gross return per $1 if the bet wins). Net odds b = o − 1.
• Return per $1 stake: R = +b if event occurs; R = −1 otherwise.

Games

1. Two fair six-sided dice are rolled. Bets may target:
   • Specific outcomes (e.g., doubles, a specific double), and
   • The sum of the dice (2–12).
2. Three fair coins are flipped. Bets target the number of heads (0, 1, 2, 3).
3. Two cards are drawn without replacement from a standard 52-card deck. Ranks are A=1, J=11, Q=12, K=13. Bets may pay based on:
   • The sum of the two ranks, and/or
   • The product of the two ranks.

Tasks

(a) For the quoted odds/payouts, quickly estimate for each available bet:

• Expected value (EV) per $1 stake;
• Variance of return per $1 stake;
• Identify all positive-EV bets.

(b) Determine whether a risk-free arbitrage exists across the available bets. If so, construct a stake allocation guaranteeing nonnegative payoff in all outcomes, with strictly positive payoff in at least one.

(c) Suppose at least one positive-EV bet exists. Propose a bankroll-sizing strategy for bankroll B over T rounds, comparing full Kelly, fractional Kelly, and fixed-fraction sizing. Choose and justify a fraction given the estimated edge and variance.

(d) Under your chosen sizing strategy, compute or approximate the probability that cumulative profit after T rounds is positive. State assumptions (e.g., independence, identical odds) and use appropriate approximations (binomial or normal/CLT). Briefly discuss accuracy vs. speed for mental estimation.