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5899Betting Kelly on the Wrong ProbabilityAn even-money coin truly wins with probability p=0.55, but you overestimate it as p=0.65 and bet the Kelly fraction implied by your estimate. What is your actual long-run expected log-growth rate per round? Compare it to the growth you would have earned betting the correct Kelly fraction, and state what the sign of your actual growth implies.概率困难数值题未尝试面试订阅5900Higher Expected Return, Lower Compounded GrowthAn even-money coin wins with probability 0.6. Trader A always stakes the fraction f A=0.10 of wealth; Trader B always stakes f B=0.40. (i) Whose stake has the higher one-round expected (arithmetic) profit? (ii) Whose wealth compounds faster over many rounds? Explain the apparent conflict.概率中等数值题未尝试免费5901Expected Rounds to Double a Kelly BankrollA gambler bets the Kelly fraction on an even-money coin with win probability p=0.6 every round, so log-wealth is a random walk with positive drift. Let G be the per-round expected log-growth (the maximal Kelly growth rate). Using an optional-stopping argument on a suitable martingale, estimate the expected number of rounds until wealth first doubles. You may ignore overshoot past the doubling level.概率困难数值题未尝试面试订阅5902Kelly Sizing with an Unknown Win ProbabilityA coin's win probability is unknown, with prior \sim Beta (2,2). You observe 7 wins and 3 losses in calibration trials, then must place one even-money bet on the next flip, choosing a fraction f of wealth to maximize the expected log-wealth after that bet. What fraction should you bet, and why is the posterior mean (rather than, say, the posterior mode) the right quantity to plug into the Kelly formula?概率中等数值题未尝试面试订阅5903Capping the Single-Bet DrawdownYou bet a fraction f of wealth on an even-money coin with win probability p=0.65, but a risk rule forbids any single losing bet from cutting your wealth by more than 20\%. What fraction should you bet, and for which win probabilities p does this drawdown rule actually constrain you below the Kelly fraction?概率简单数值题未尝试免费5904Kelly Exceeds Full InvestmentA favorable bet has limited downside: staking a fraction f of wealth, you gain the full amount f with probability p=0.7 but lose only half the stake, 0.5f, with probability 0.3. (a) Find the growth-optimal fraction f *. (b) If you cannot borrow (so f\le 1, i.e. you can stake at most all your wealth), what fraction do you actually bet?概率简单数值题未尝试免费5905Kelly with a Cash-Reserve FloorYou may stake on an even-money coin with win probability p=0.8, but a liquidity rule requires you to keep at least half your total wealth in untouched cash at all times, so the staked fraction satisfies f\le 0.5. Set up the constrained maximization of expected log-growth, use the KKT conditions to determine the optimal stake, and state whether the reserve constraint binds.概率困难数值题未尝试面试订阅5906How Many Bets Until Loss Is UnlikelyA Kelly bettor on an even-money coin with p=0.6 stakes the optimal fraction f *=0.2 each round. The per-round log-return is +\ln 1.2 with probability 0.6 and \ln 0.8 with probability 0.4, with mean G\approx0.0201 and variance v\approx0.0395. Using Chebyshev's inequality, find a number of rounds n after which the probability of ending below the starting wealth is at most 5\%.概率困难数值题未尝试面试订阅5907Kelly with a Proportional Trading CostOn an even-money coin with win probability p, each round you pay a proportional cost c on the amount staked, regardless of the outcome. So staking fraction f, a win multiplies wealth by 1+f(1-c) and a loss by 1-f(1+c). Derive the growth-optimal fraction f * in terms of p and c, evaluate it for p=0.6,\ c=0.05, and find the cost level at which the optimal stake drops to zero.概率困难数值题未尝试面试订阅5908Reaching the Target Against a House EdgeYou hold 3 chips and bet one chip per round on an even-money game that you win with probability p=0.4 (and lose with probability 0.6). You keep playing until you either reach 5 chips (cash out) or hit 0 (broke). What is the probability you reach 5 chips before going broke?概率中等数值题未尝试免费5909Bold Play to a Quadrupling GoalYou have \2 and want to reach \8 on an even-money game you win with probability p=0.4. You use bold play: each round you stake the most that keeps you from overshooting the goal, i.e. \min( current wealth ,\ 8- current wealth ). What is the probability bold play reaches the \8 goal?概率中等数值题未尝试免费5910The Martingale Doubling System on RouletteOn a roulette red bet you win (even money) with probability 18/38 and lose with probability 20/38. You run the doubling (martingale) system aiming to win \1: bet \1; if it loses, bet \2; if that loses, bet \4. You stop after a win or after three straight losses (your bankroll of \7 is then gone). Find (a) the probability the campaign ends in ruin and (b) the expected net profit of the campaign.概率中等数值题未尝试面试订阅5911How Long Can You Play Before the Edge Eats YouYou start with \2 and bet \1 per round on an even-money game you win with probability p=0.4. You play until you either reach \5 or go broke. What is the expected number of rounds you play before the game ends?概率困难数值题未尝试面试订阅5912All-In or Split the Stake to Double UpYou have \1 and want to double it to \2 on a subfair even-money bet that wins with probability p=0.45 (and you quit forever once you either reach \2 or hit \0). Compare two plans: (A) bet the whole \1 in one shot; (B) bet \0.50 each round until you reach \2 or go broke. Which gives the higher probability of doubling, and what are the two probabilities?概率困难数值题未尝试面试订阅5913Minimizing Ruin in a Favorable GameYou have 2 chips and play a favorable even-money game you win with probability p=0.6, intending to play forever (no cash-out target) and stake whole chips. To minimize the chance of ever going broke you bet the smallest stake, 1 chip per round. What is the probability you are eventually ruined under this minimum-stake (timid) play?概率中等数值题未尝试免费5914Red-and-Black Bold Play from Three-QuartersIn red-and-black you bet on an even-money outcome that comes up with probability p=0.4, scaling all amounts so the goal is \1. You currently hold \0.75 and use bold play: stake \min( current ,\ 1- current ) each round, trying to reach \1 before reaching \0. What is the probability bold play reaches the goal?概率中等数值题未尝试面试订阅5915Timid Versus Bold to QuadrupleStarting with \1 you want to reach \4 on an even-money game you win with probability p=0.4, quitting when you reach \4 or go broke. Compute the probability of reaching \4 under (A) timid play, betting \1 each round, and (B) bold play, staking \min( current ,\ 4- current ). Which strategy gives the higher chance of reaching the goal?概率困难数值题未尝试面试订阅5916Most You Would Pay for a Perfect TestA product launch pays +30 if the market is receptive and -12 if it is not; receptivity has prior probability 3 10 . You may instead shelve the product for 0. A consultant offers a perfectly accurate test that reveals the true market state before you decide. What is the most you should be willing to pay for this test?概率中等derivation未尝试免费5917Free Peek Before Calling the Bigger BoxTwo boxes each independently contain an amount drawn uniformly from \ 1,2,3,4\ . You must guess which box holds the strictly larger amount; a correct guess pays 1 and a tie or wrong guess pays 0. Before guessing you may take a free peek at the contents of one box (your choice of which). By how much does this peek increase your probability of a correct guess compared with guessing blind?概率简单derivation未尝试免费5918Defective-Batch Inspection With an Imperfect DetectorA batch is defective with prior probability \frac14. Accepting a good batch pays +20; accepting a defective batch pays -40; rejecting pays 0. Before deciding you may run a detector that flags 'defective.' It flags a truly defective batch with probability 9 10 and a good batch with probability \frac15 (false positive). What is the value of running the detector (the increase in expected payoff from using it optimally)?概率困难derivation未尝试面试订阅