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5919One Free Draw Before Betting on the Majority ColorAn urn is type-R with probability \frac35 (then it is 80\% red balls) or type-B with probability \frac25 (then it is 80\% blue balls). You will bet on the urn's majority color: a correct bet pays 1, a wrong bet pays 0. You may first draw one ball (with replacement) and observe its color for free. By how much does observing this single draw raise your expected payoff over betting with no draw?概率中等derivation未尝试免费5920Clairvoyance Across Three StatesThe state is High, Mid, or Low with probabilities \frac12,\frac13,\frac16. You pick action Long or Flat once. Long pays 12,\ -3,\ -9 in High, Mid, Low respectively; Flat pays 0 in every state. A clairvoyant will tell you the exact state before you choose. What is the difference between your expected payoff acting on the clairvoyant's report and your expected payoff using the single best action chosen in advance?概率中等derivation未尝试免费5921Is the Analyst's Report Worth Its FeeAn investment pays +14 if a deal closes and -10 if it falls through; closing has prior probability \frac12. You may invest or pass (pass pays 0). For a fee of 2 you may buy an analyst report that correctly predicts the outcome with probability 7 10 , after which you decide. Should you buy the report, and what is its value net of the no-report optimum?概率中等derivation未尝试免费5922Three-Candidate Best-ChoiceThree candidates of distinct, unknown qualities arrive in uniformly random order. After each interview you learn only the candidate's rank relative to those seen so far, and you must immediately and irrevocably hire or reject. You want to maximize the probability of hiring the single best candidate. What is the optimal policy and the resulting probability of success?概率简单数值题未尝试免费5923Full-Information Uniform StoppingYou observe up to three independent draws from the Uniform(0,1) distribution, one at a time, and after each you may stop and collect the value just seen or discard it and continue (no recall of discarded values). If you reach the third draw you must take it. Knowing the distribution exactly, what stopping policy maximizes the expected value collected, and what is that expected value?概率简单数值题未尝试免费5924House-Selling Stationary ThresholdOffers arrive sequentially and independently, each Uniform(0,1). After each offer you either accept it (and stop) or reject it forever (no recall) and wait for the next, paying a fixed search cost c per rejected offer. There is no deadline. For c = 0.02, find the optimal stationary acceptance threshold and the expected net payoff under it.概率中等数值题未尝试免费5925Prophet vs Gambler, Two PrizesTwo prizes arrive in sequence, each an independent Uniform(0,1) value, revealed one at a time; you must accept one immediately when shown (no recall, and if you reject the first you must take the second). A prophet who sees both in advance collects E[max(X1,X2)]. Compute the prophet's expected reward, the best the online gambler can guarantee in expectation, and the ratio of gambler to prophet.概率简单数值题未尝试免费5926Settling For Top TwoFour items arrive in uniformly random order; you observe only relative ranks and must accept one irrevocably (if you reach the last item you take it). Unlike the classic problem, you win if the item you accept is either the best OR the second-best overall. Find the policy that maximizes your winning probability and that maximum probability.概率困难数值题未尝试面试订阅5927Minimize the Expected RankThree items arrive in random order; after each you learn its rank relative to those seen so far and must accept or reject irrevocably (the last is forced). Instead of trying to get the single best, you want to minimize the expected absolute rank of the item you finally accept (rank 1 = best, rank 3 = worst). Find the optimal policy and the minimum achievable expected rank.概率中等数值题未尝试免费5928Two Hiring AttemptsFour candidates arrive in random order with only relative ranks observable. You are allowed to make TWO irrevocable picks during the stream (you point at a candidate, that candidate is chosen, then later you may point at one more); you win if AT LEAST ONE of your two chosen candidates is the overall best. Choices are made online with no recall. What is the optimal strategy and the probability of winning?概率困难数值题未尝试面试订阅5929Pay-Per-Look With RecallYou may sample as many independent Uniform(0,1) values as you like, paying a cost c per sample. Recall is allowed, so at any time you may stop and collect the maximum value seen so far. With c = 0.125, find the optimal stopping rule (the reservation level r above which you stop) and the expected net payoff (max value collected minus total sampling cost).概率中等数值题未尝试免费5930Why Full Information Beats RanksTwo candidates have qualities drawn iid from a KNOWN Uniform(0,1) distribution, revealed as actual numerical values one at a time; you must accept one (reject-the-first forces the second). You win only if you accept the candidate with the higher quality. Compare the best win probability achievable by a full-information policy that uses the numerical value of candidate 1 against the best win probability of a rank-only policy that sees only relative order. Which is larger and by how much?概率简单derivation未尝试免费5931Discounted Offer StoppingEach period an offer arrives, iid Uniform(0,1). If you accept an offer of value x at period t, you receive beta t * x, where beta = 0.9 is a per-period discount factor (so waiting shrinks the value of any future acceptance). No recall, infinite horizon. Find the optimal stationary acceptance threshold and the expected discounted payoff from the start.概率中等数值题未尝试免费5932Catch the Last SuccessFive deals appear in sequence. Independently, each deal turns out to be 'live' with probability 0.2 (and 'dead' otherwise); you learn live/dead immediately when the deal appears and must accept or pass irrevocably with no recall. You win precisely if you accept the LAST live deal of the five. Using the odds-algorithm logic for such problems (sum the odds r i = p i/(1-p i) from the end until the running total first reaches 1, and start accepting any live deal from that index onward), find the optimal stopping index and your probability of winning.概率困难数值题未尝试面试订阅5933The Postdoc Problem (Pick Second-Best)Four applicants of distinct unknown qualities arrive in uniformly random order; after each you learn only its rank relative to those seen so far and must irrevocably accept or reject (if you reach the last you must take it). A famous twist: you win only if the applicant you accept is the SECOND-best of all four (the very best is taken by a rival institution, so picking the best is a loss). Find the optimal policy and the maximum probability of landing exactly the second-best.概率困难数值题未尝试面试订阅5934Sultan's Dowry with Five SuitorsA sultan offers a vizier the hand of one of 5 daughters, presented one at a time in uniformly random order. After meeting a daughter, the vizier learns only how she ranks (by dowry) relative to those already seen and must immediately and irrevocably accept or send her away; if he reaches the last he must take her. He wins only by choosing the single highest dowry. Among all 'skip the first r, then take the first daughter who beats every earlier one' rules, find the optimal r and the exact probability of winning.概率中等数值题未尝试免费5935Hiring With an Interview CostThree candidates of distinct unknown qualities arrive in uniformly random order; each candidate must be interviewed (in order) before you can judge their relative rank, and conducting each interview costs 0.05 utility. After interviewing a candidate you immediately and irrevocably hire or reject (if you reach the third you must hire). You receive payoff 1 if you hire the overall best candidate and 0 otherwise, minus the total interview cost incurred. Among the threshold rules 'reject the first r interviewed, then hire the first later record', find the optimal r and the resulting expected net payoff.概率中等数值题未尝试免费5936Sell Before the Deadline in a Falling MarketYou must sell an asset within 3 periods. In period t one offer arrives, drawn uniformly on [0, 1 - 0.2*(t-1)] (a declining market: the range is [0,1] in period 1, [0,0.8] in period 2, [0,0.6] in period 3). You accept and stop, or reject forever (no recall). If you reach period 3 you must accept that offer. Knowing these distributions, find the optimal acceptance thresholds and the expected sale price under the optimal policy.概率中等数值题未尝试免费5937Reservation Wage With Exponential OffersJob offers arrive sequentially and independently, each an Exponential random variable with rate 1 (mean 1). After each offer you accept it (and stop) or reject it forever and pay a search cost c = 0.2 to see the next; there is no deadline. Find the optimal stationary reservation level a above which you accept, and the expected wage you end up accepting under that policy.概率中等数值题未尝试免费5938Bird in the Hand vs Discounted WaitYou face two periods. In period 1 a reward X1 ~ Uniform(0,1) is offered; accept it now to receive X1, or wait. If you wait, in period 2 you must accept X2 ~ Uniform(0,1), but a reward received in period 2 is worth only a fraction beta = 0.8 of its face value (discounting). No recall. Find the optimal period-1 acceptance threshold and the expected payoff of the optimal policy.概率简单数值题未尝试免费