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5939Secretary Selection When Ties Are PossibleThree items arrive in uniformly random order. Their qualities are NOT all distinct: two of them have quality 2 (tied for best) and one has quality 1. After each item you observe its quality relative to those seen so far, reported as 'higher', 'tied', or 'lower' (so a tie is visible). You accept irrevocably or reject (the last is forced). You want to maximize the expected quality of the item you accept. Find the optimal policy and the maximum expected quality, and explain how the possibility of an observed tie changes what you can guarantee.概率中等数值题未尝试免费5940Secretary With an Unknown Number of CandidatesCandidates arrive one at a time in uniformly random order, but the TOTAL number N is itself random: N = 2 with probability 1/2 and N = 3 with probability 1/2, and you do not learn N in advance. After each arriving candidate you observe its rank relative to those seen so far and must irrevocably accept or pass; once the stream ends, if you never accepted you lose. You win only if the candidate you accept is the overall best of all N who arrived. Find the policy that maximizes the win probability and that probability.概率困难数值题未尝试面试订阅5941The 1/e Law of Best ChoiceIn the classic secretary problem with n candidates (relative ranks only, irrevocable choices), the look-then-leap rule observes the first r candidates without choosing and then accepts the first later candidate who beats all seen so far. For large n, write r = t*n and derive the limiting win probability as a function of the skip fraction t in (0,1). Then find the t that maximizes it and the resulting optimal asymptotic probability of selecting the single best candidate.概率中等derivation未尝试免费5942Collecting a Full Set of SixA vending machine dispenses one of 6 distinct toy types, each equally likely and independent across purchases. What is the expected number of purchases needed until you own at least one of every type?概率中等数值题未尝试免费5943Variance of Completing a Four-Sticker SetStickers come in 4 equally likely types, independent across packs. Let T be the number of packs you open until you have collected all 4 types. Compute Var (T).概率困难数值题未尝试面试订阅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