Analyze EV, arbitrage, and bet sizing across games

You face three independent betting games: ( 1) roll two fair six-sided dice; bets may target specific outcomes (e.g., doubles) and the sum of the dice; ( 2) flip three fair coins; bets target the number of heads; ( 3) draw two cards without replacement from a standard 52-card deck, with ranks A=1, J=11, Q=12, K=13; bets pay based on the sum and on the product of the two ranks. For any quoted odds or payout tables you are given, do the following: (a) quickly estimate each bet’s expected value and variance and identify all positive-EV bets; (b) determine whether a risk-free arbitrage exists across the available bets and, if so, construct the allocation that guarantees nonnegative payoff in all outcomes with strictly positive payoff in at least one; (c) propose a bet-sizing strategy for a bankroll B over T rounds when at least one positive-EV bet exists—compare full Kelly, fractional Kelly, and fixed-fraction sizing, and justify how you choose the fraction given edge and variance; (d) under your chosen sizing strategy, compute or approximate the probability that cumulative profit after T rounds is positive, stating assumptions (e.g., independence, identical odds) and using appropriate approximations (e.g., binomial or normal/CLT), and discuss accuracy versus speed for mental estimation.