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029Posterior After One Correct and One Incorrect ForecastAn analyst is senior with prior probability 0.3 and junior otherwise. A senior analyst calls a binary market move correctly with probability 0.8 on each day; a junior does so with probability 0.6. On two independent days you observe exactly one correct call and one incorrect call from the same analyst. What is the posterior probability the analyst is senior?概率中等数值题未尝试免费030Three Suppliers and a Two-Stage InspectionA firm sources components from three suppliers: S 1 (50% of supply, 2% defect rate), S 2 (30%, 3% defect rate), S 3 (20%, 5% defect rate). A component is selected at random and undergoes two independent inspection stages. Stage 1 detects a defect with probability 0.8 (if defective) and falsely flags a good component with probability 0.05. Stage 2 detects a defect with probability 0.9 (if defective) and falsely flags with probability 0.03. A component is flagged by both stages. Compute: (a) P( component is actually defective \mid flagged by both stages ). (b) Given that the component is actually defective and flagged by both stages, what is the probability it came from S 3?概率困难数值题未尝试面试订阅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032Three-Source Posterior After Alert and Failed ClearanceA transaction comes from source A, B, or C with priors 1/2, 1/3, and 1/6. The alert probabilities are 0.2, 0.4, and 0.8 respectively. Conditional on being alerted, the clearance probabilities are 0.9, 0.6, and 0.25 respectively. Given that the transaction was alerted and then failed clearance, what is the posterior probability it came from source C?概率中等数值题未尝试免费033Three-Regime Bayesian Updating from Daily ReturnsA portfolio manager models the market as being in one of three regimes, each equally likely a priori: - **Bull**: the stock goes up on any given day with probability 4 5 . - **Neutral**: the stock goes up with probability 1 2 . - **Bear**: the stock goes up with probability 1 5 . Days are conditionally independent given the regime. Over three days the stock goes up, up, then down. (a) What is the posterior probability of each regime? (b) What is the conditional probability the stock goes up on Day 4?概率中等数值题未尝试免费034Repeated Positive Test PosteriorA rare condition has prior probability 1/100. Each test is independent conditional on the true state, with true-positive rate 19/20 and false-positive rate 1/10. If two tests are both positive, what is the posterior probability of the condition?概率困难数值题未尝试面试订阅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037Two Correct Calls PosteriorAn analyst is high-skill with prior probability 2/5. A high-skill analyst makes a correct call with probability 4/5; a low-skill analyst does so with probability 11/20. If two independent calls are both correct, what is the posterior probability the analyst is high-skill?概率简单数值题未尝试免费039Double Alert PosteriorA transaction is malicious with prior probability 1/40. Two independent alert engines each fire with probability 7/10 on malicious transactions and 1/20 on benign ones. If both engines fire, what is the posterior malicious probability?概率中等数值题未尝试免费040Sequential Signal Updating and the Tower PropertyA quant researcher believes a directional signal has accuracy p that is either 1 3 or 2 3 , each equally likely a priori. On each day the signal independently (given p) predicts the market direction, and is correct with probability p. (a) On Day 1 the signal is correct. What is the posterior P\! (p = \tfrac 2 3 \mid C 1 )? (b) On Day 2 the signal is wrong. Starting from the Day-1 posterior, compute the updated P\! (p = \tfrac 2 3 \mid C 1, W 2 ). (c) Verify the tower property of conditional expectation: show that E[p] = E\! [\,E[p \mid D 1]\, ], where D 1 \in \ C 1, W 1\ denotes the Day-1 outcome. Compute all quantities explicitly.概率困难derivation未尝试面试订阅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043Factory Defect Tracing and Predictive InferenceA factory has three production lines with the following output shares and defect rates: | Line | Share of output | Defect rate | |------|----------------|-------------| | 1 | 50% | 2% | | 2 | 30% | 3% | | 3 | 20% | 5% | An item is selected at random from today's output and found to be defective. (a) What is the posterior probability that the item came from each production line? (b) A second item is now drawn independently from the same (unknown) production line. What is the conditional probability that this second item is also defective, given that the first was defective?概率中等数值题未尝试免费045Multi-Analyst Signal Fusion and Sequential UpdatingA trading desk receives directional predictions from three independent analysts. The market will either go "up" or "down", each with prior probability 1 2 . Each analyst independently (given the true state) predicts the correct direction with the following probabilities: - Analyst 1: accuracy 0.8 - Analyst 2: accuracy 0.7 - Analyst 3: accuracy 0.9 (a) All three analysts predict "up". What is P( up \mid all three say up )? (b) Analysts 1 and 2 predict "up", but Analyst 3 predicts "down". What is P( up \mid U 1, U 2, D 3)? (c) Show that the posterior in part (b) can be computed by sequential Bayesian updating — updating on one analyst's signal at a time — and that the final answer does not depend on the order of updates. Demonstrate this by computing the posterior via two different orderings.概率困难derivation未尝试面试订阅049Mystery Coin IdentificationA box contains three coins, each equally likely to be selected: - **Coin F** (fair): P(H) = 1 2 - **Coin H** (heads-biased): P(H) = 3 4 - **Coin T** (tails-biased): P(H) = 1 4 You pick one coin uniformly at random and flip it three times, obtaining the sequence H, T, H (flips are conditionally independent given the coin). (a) Find the posterior probability that the coin is each type. (b) What is the conditional probability that the fourth flip is heads?概率困难数值题未尝试免费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051Pairwise Independence Check for Dice EventsRoll two fair six-sided dice. Define events A = \ sum is 7\ and B = \ first die shows an even number \ . Are A and B independent? Prove your answer by computing P(A), P(B), and P(A \cap B).概率简单数值题未尝试免费052Pairwise but Not Mutual IndependenceLet the sample space be \Omega = \ 1, 2, 3, 4\ with uniform probability P(\ i\ ) = 1/4. Define A = \ 1, 2\ , B = \ 1, 3\ , C = \ 1, 4\ . Show that A, B, C are pairwise independent but not mutually independent.概率简单derivation未尝试免费053Zero Correlation Does Not Imply IndependenceLet X be uniformly distributed on \ -1, 0, 1\ . Define Y = X 2. Compute Cov (X, Y) and determine whether X and Y are independent.概率中等derivation未尝试免费054Conditional Independence Breaks Under MarginalizationA coin is chosen at random: with probability 1/2 it is fair (p = 1/2) and with probability 1/2 it is biased (p = 1, always heads). Let A be the event that the first flip is heads and B the event that the second flip is heads. Show that A and B are conditionally independent given the coin type C, but that A and B are not marginally independent. Compute P(B \mid A).概率中等derivation未尝试免费055Independent Events Become Dependent Under ConditioningLet A and B be independent events with P(A) = P(B) = 1/2. Define D = A \triangle B (exactly one of A, B occurs). (a) Compute P(D), P(A \mid D), and P(A \cap B \mid D). (b) Are A and B conditionally independent given D? (c) Are A and B conditionally independent given D c?概率困难derivation未尝试面试订阅