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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