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5944Two Unequal PrizesEach cereal box contains one prize: a common prize with probability 2/3 and a rare prize with probability 1/3, independent across boxes. What is the expected number of boxes you must open to have collected both prizes?概率中等数值题未尝试免费5945Two of EverythingA machine outputs one of 3 equally likely tokens per play, independent across plays. What is the expected number of plays until you hold at least TWO copies of each of the 3 token types?概率困难数值题未尝试面试订阅5946Halfway CollectionTrading cards come in 10 equally likely types, independent across packs. What is the expected number of packs needed until you own 5 distinct types (any 5, not a specific set)?概率中等数值题未尝试免费5947Two Stickers to GoYou are collecting 6 equally likely sticker types (independent packs). You currently hold exactly 4 distinct types. What is the expected number of ADDITIONAL packs needed to complete the set of 6?概率简单数值题未尝试免费5948Empty Mailboxes10 letters are placed independently and uniformly at random into 8 mailboxes. What is the expected number of mailboxes that remain empty?概率简单数值题未尝试免费5949Last Card in the DeckYou collect cards drawn uniformly and independently from 5 equally likely types until you own all 5. What is the probability that a specified type (say type A) is the LAST one you complete the set with?概率中等数值题未尝试免费5950Variance of the Coverage Count4 balls are thrown independently and uniformly into 6 boxes. Let D be the number of boxes that receive at least one ball. Compute Var (D).概率困难数值题未尝试面试订阅5951Three Prizes, Unequal OddsA claw machine yields prize 1 with probability 1/2, prize 2 with probability 1/3, and prize 3 with probability 1/6, independently each play. What is the expected number of plays to collect all three prizes?概率困难数值题未尝试面试订阅5952Distinct Types Across Two PacksA booster pack contains 4 cards drawn WITHOUT replacement from a pool of 9 equally likely distinct types (so the 4 cards in a single pack are all different types). You open two packs; the two packs are independent of each other (8 cards total). What is the expected number of DISTINCT types you own?概率简单数值题未尝试免费5953Collecting in PairsEach purchase gives you a PACK of 2 cards, where each of the 2 cards is independently and uniformly one of 4 types (the two cards in a pack may coincide). What is the expected number of PACKS you must buy to collect all 4 types?概率困难数值题未尝试面试订阅5954Just These TwoCoupons arrive uniformly and independently from 5 types. You only care about two SPECIFIC types, the gold and the silver coupon; the other three are worthless to you. What is the expected number of draws until you have collected BOTH the gold and the silver coupon?概率中等数值题未尝试免费5955Buy Random or Buy the Missing OneYou need all 5 types and currently hold 4 distinct types (exactly one type missing). Each round you may either (a) buy a random coupon for 1 (uniform over all 5 types), or (b) directly buy your missing type from a reseller for 5. Acting optimally to minimize expected total future cost, what is your minimum expected cost to complete the set?概率中等数值题未尝试免费5956How Much by the DeadlineYou will draw exactly 6 coupons, each uniform and independent over 4 types. The promotion ends after these 6 draws. What is the expected number of DISTINCT types you will have collected by the deadline?概率简单数值题未尝试免费5957The Busiest Bin3 balls are thrown independently and uniformly into 3 bins. Let M be the maximum load, i.e. the number of balls in the most-loaded bin. What is E[M]?概率中等数值题未尝试免费5958Bins With Exactly Two9 balls are thrown independently and uniformly into 6 bins. What is the expected number of bins that contain EXACTLY 2 balls?概率简单数值题未尝试免费5959Coupons With BlanksEach draw is a blank with probability 1/4 (no coupon), and otherwise (probability 3/4) yields one of 3 collectible types, each equally likely. What is the expected number of draws to collect all 3 types?概率中等数值题未尝试免费5960Every Bin Gets a PairBalls are dropped one at a time, each independently into one of 2 equally likely bins. What is the expected number of balls dropped until EVERY bin contains at least 2 balls?概率中等数值题未尝试免费5961A Useful TradeTwo collectors each independently draw a single coupon, uniform over 4 types. They are willing to TRADE only if a swap benefits both, which happens exactly when their two coupons are of different types. What is the probability that a beneficial trade is possible?概率简单数值题未尝试免费5962Time to Climb One Step (Biased Walk)A walk starts at 0 and each step moves +1 with probability 2/3 and -1 with probability 1/3. Let T be the first time it reaches +1. Find E[T].概率中等数值题未尝试免费5963Wald's Identity from Optional StoppingLet X 1,X 2,... be i.i.d. with mean 4, and let N be a stopping time (with respect to the X's) with E[N]=10. Using the martingale M n = sum i<=n X i - 4n and optional stopping, find E[X 1+...+X N].概率简单数值题未尝试免费