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027Prior Needed to Break Even at 50-50 PosteriorA signal has likelihood ratio 5 in favor of the hypothesis H relative to not-H. What prior probability p of H makes the posterior exactly 1/2 after seeing the signal?概率简单数值题未尝试免费031Flagged but Then ClearedA borrower is high risk with prior probability 0.2. A fast model flags a high-risk borrower with probability 0.7 and flags a low-risk borrower with probability 0.1. Only flagged borrowers get a manual review. A high-risk flagged borrower passes manual review with probability 0.4, while a low-risk flagged borrower passes it with probability 0.7. Given that a borrower was flagged and then passed manual review, what is the posterior probability the borrower is high risk?概率简单数值题未尝试免费035Positive Then Negative PosteriorA model-risk event has prior probability 1/50. Conditional on the event, the probabilities of a positive screen and then a negative manual review are 9/10 and 1/10. Conditional on no event, those probabilities are 1/5 and 4/5. If the observed sequence is positive then negative, what is the posterior event probability?概率困难数值题未尝试面试订阅041Two Screening Passes PosteriorA candidate belongs to the top tier with prior probability 1/4. A top-tier candidate passes each screening round with probability 9/10; a non-top-tier candidate passes with probability 3/5. If the candidate passes two independent rounds, what is the posterior top-tier probability?概率简单数值题未尝试免费042Prior Recovery from a PosteriorA hypothesis has prior probability p. An observed signal has likelihood 3/4 under the hypothesis and 1/4 under the alternative. After observing the signal, the posterior becomes 3/7. What was p?概率中等数值题未尝试免费050Noisy Signal Detection and Evidence ThresholdA hidden signal S is equally likely to be +1 or -1. At each time step you receive a noisy reading: if S = +1 the reading is +1 with probability 2 3 and -1 with probability 1 3 ; if S = -1 the reading is -1 with probability 2 3 and +1 with probability 1 3 . Readings are conditionally independent given S. (a) You observe the sequence (+1, +1, -1). Find the posterior probability P(S = +1 \mid observations ). (b) Starting from the uniform prior, what is the minimum number n of consecutive +1 readings required so that P(S = +1 \mid n consecutive +1) > 0.95?概率困难数值题未尝试免费078Simpson's Paradox in a Clinical TrialA clinical trial tests a drug on two subgroups. In subgroup A (mild cases): 81/87 (93%) of treated patients recover vs 234/270 (87%) of untreated. In subgroup B (severe cases): 192/263 (73%) of treated patients recover vs 55/80 (69%) of untreated. The drug improves recovery rates in both subgroups. Now compute the overall recovery rates (combining subgroups) for treated vs untreated. Explain the apparent contradiction and identify the lurking variable that causes it.概率中等derivation未尝试免费080The Two-Envelope ParadoxTwo envelopes each contain a positive amount of money; one contains exactly twice the other. You pick one envelope at random and find it contains x dollars. The naive argument says: the other envelope is equally likely to contain 2x or x/2, so the expected value of switching is (1/2)(2x) + (1/2)(x/2) = 5x/4 > x, and you should always switch — but this leads to the absurd conclusion that you should switch back and forth indefinitely. (a) Identify the precise flaw in the naive argument. (b) Suppose the smaller amount S is drawn from a known proper prior distribution with E[S] = < . Show that the unconditional expected gain from switching is zero. (c) Explain why conditional on observing x, it *can* be rational to switch for some values of x and not others.概率困难derivation未尝试面试订阅083The Necktie ParadoxTwo players, Alice and Bob, each receive a necktie as a gift. The prices of the two neckties are different positive values. Neither player knows either price. They agree to compare: whoever has the cheaper necktie wins the other's necktie. Alice reasons: 'If my necktie costs x, then I either gain a necktie worth more than x or lose a necktie worth x. Since each is equally likely, my expected gain from the game is positive.' Bob makes the identical argument. Both conclude the game favors them — a contradiction since this is a zero-sum exchange. (a) Identify the precise flaw in Alice's reasoning. (b) Suppose the two necktie prices are drawn by a third party as V and 2V where V \sim Uniform (1, 100), assigned randomly to Alice and Bob. If Alice sees her necktie has price tag x, what is her expected gain from the game as a function of x? Show that the unconditional expected gain is zero.概率中等derivation未尝试免费085The Inspection Paradox (Bus Waiting Time)Buses arrive at a stop according to a Poisson process with rate (so inter-arrival times are iid Exp ( ) with mean 1/ ). You arrive at the bus stop at a uniformly random time, independent of the bus schedule. Let L be the length of the inter-arrival interval that contains your arrival time — i.e., the time between the last bus before you arrived and the next bus after. (a) Find E[L]. Explain why it is **not** 1/ despite inter-arrival times having mean 1/ . (b) Find the expected waiting time E[W] until the next bus, where W is the time from your arrival until the next bus. (c) A city official surveys bus riders and asks how long they waited. If the reported average is 1/ , should the transit authority be surprised? Explain using the inspection paradox.概率困难derivation未尝试面试订阅088The St. Petersburg ParadoxA casino offers the following game. A fair coin is flipped repeatedly until the first Tails appears. If the first Tails occurs on flip n, you win 2 n dollars. (a) Compute the expected payoff of the game. (b) Despite the answer to (a), most people would pay no more than about \20 to play. Resolve this apparent paradox using Daniel Bernoulli's approach: assume the player has log utility u(x) = \ln(x) and initial wealth W. Compute the expected utility of the game and find the certainty equivalent (the sure amount that gives the same expected utility) for W = 1 , 000 , 000. (c) A more practical resolution: suppose the casino has finite total capital C. If the payout is capped at C = 2 40 (about \1 trillion), what is the expected payoff?概率中等derivation未尝试免费110The 4-3-3-3 Bridge Hand DistributionA 13-card bridge hand is dealt from a standard 52-card deck. What is the probability that the hand has a 4-3-3-3 suit distribution (exactly four cards of one suit and exactly three cards of each of the other three suits)?概率困难derivation未尝试面试订阅115Void in a Bridge HandA 13-card bridge hand is dealt from a standard 52-card deck. What is the probability that the hand contains a void — that is, at least one suit is completely absent from the hand?概率困难derivation未尝试面试订阅120Expected Distinct Ranks in a Seven-Card HandSeven cards are drawn without replacement from a standard 52-card deck. Let R be the number of distinct ranks represented among the seven cards. Find E[R].概率困难derivation未尝试面试订阅125All Four Suits in a Seven-Card HandSeven cards are drawn without replacement from a standard 52-card deck. What is the probability that all four suits are represented among the seven cards?概率困难derivation未尝试面试订阅165Expected Number of People Who Share Someone's BirthdayFor n independent uniform birthdays on a 365-day calendar, what is the expected number of people whose birthday is shared by at least one other person?概率困难derivation未尝试面试订阅168Pair Expectation Under a Nonuniform CalendarBirthdays fall on days 1 through m with probabilities p1,...,pm, not necessarily uniformly. For n independent people, what is the expected number of unordered matching pairs?概率中等derivation未尝试面试订阅195Expected Time to First Collision in Six UrnsBalls are thrown one at a time, each landing independently and uniformly at random into one of 6 urns. Let T be the index of the first ball that lands in an already-occupied urn (so T \ge 2). Derive E[T] and give an exact fraction.概率困难derivation未尝试免费210Multinomial Covariance and Conditional DistributionA fair die is rolled n = 60 times. Let X i be the number of times face i appears, for i = 1, \ldots, 6, so (X 1, \ldots, X 6) \sim Multinomial (60,\, 1/6, \ldots, 1/6). (a) Using indicator variables, compute Cov (X 1, X 2). (b) Find the correlation (X 1, X 2). (c) Determine the conditional distribution of (X 2, X 3, X 4, X 5, X 6) given X 1 = 12. What is E[X 2 \mid X 1 = 12]?概率困难derivation未尝试免费214Poisson Thinning and Independence of Split StreamsLet N \sim Poisson ( ). Each of the N events is independently classified as type 1 with probability p and type 2 with probability 1 - p. Let N 1 and N 2 denote the counts of type 1 and type 2 events, respectively. (a) Derive the marginal distribution of N 1. (b) Derive the joint PMF P(N 1 = j, N 2 = k) and show that N 1 and N 2 are independent. (c) A website receives page views at rate = 200 per hour. Each visitor independently converts (makes a purchase) with probability p = 0.03. Find the probability of exactly 4 conversions in an hour, and the probability of at least 1 conversion given at most 210 total page views.概率困难derivation未尝试免费