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026Conflicting Two-Screen UpdateA loan is distressed with prior probability 0.3. Screen A flags a distressed loan with probability 0.8 and flags a healthy loan with probability 0.1. Any flagged loan then goes to screen B, which passes a distressed loan with probability 0.25 and passes a healthy loan with probability 0.70. Given that A flagged the loan and B did not pass it, what is the posterior probability the loan is distressed?概率简单数值题未尝试免费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028Factory Posterior After One Good and One Bad ItemA part comes from factory A with prior probability 0.6 and from factory B otherwise. Factory A produces defects with probability 0.1; factory B produces defects with probability 0.4. Two parts from the same unknown factory are inspected and exactly one is defective. What is the posterior probability the parts came from factory A?概率中等数值题未尝试免费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038Intraday Momentum and Regime ClassificationA quantitative analyst classifies each trading day as either "trending" (probability 0.6) or "mean-reverting" (probability 0.4). - On a trending day, the morning session is positive with probability 0.7, and given a positive morning the afternoon session is also positive with probability 0.8. - On a mean-reverting day, the morning session is positive with probability 0.5, and given a positive morning the afternoon session is positive with probability 0.3. Today both the morning and afternoon sessions are positive. What is the posterior probability that today is a trending day?概率中等数值题未尝试免费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044Posterior Odds UpdateThe prior probability of a hypothesis is 3/10. An observed signal has likelihood ratio (4/5)/(2/5). What are the posterior odds in favor of the hypothesis?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046Two Red Draws PosteriorA hidden urn is equally likely to be A or B. Urn A gives red with probability 4/5 and urn B gives red with probability 1/3. Two independent draws with replacement are both red. What is the posterior probability the urn is A?概率简单数值题未尝试免费