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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