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3166Signal Before a Binary TradeA trade pays +8 in a favorable state and -5 in an unfavorable state. The favorable state has prior probability 2 5 . Before trading, you may buy a signal for cost 1 2 ; it is correct with probability 4 5 . If you see the signal, you may either trade or abstain after observing it. What is the value of the signal, and should you buy it at that cost?概率中等derivation未尝试面试订阅3176Aggressive vs Defensive Quote After a SignalThere are two possible actions. `Aggressive` pays 10 in the good state and -8 in the bad state. `Defensive` pays 4 in the good state and -1 in the bad state. The good state has prior probability 2 5 . Before acting, you may see a binary signal that is correct with probability 4 5 . What is the value of observing the signal, and which action should you take after a good signal and after a bad signal?概率困难derivation未尝试面试订阅3177Allocate Between Fast and Safe BooksThere are two possible actions. `Aggressive` pays 9 in the good state and -6 in the bad state. `Defensive` pays 5 in the good state and 1 in the bad state. The good state has prior probability 1 2 . Before acting, you may see a binary signal that is correct with probability 3 4 . What is the value of observing the signal, and which action should you take after a good signal and after a bad signal?概率困难derivation未尝试面试订阅3186Perfect Information Before Choosing a Desk StrategyTwo actions are available before the state is revealed. Action A pays 10 in the good state and -4 in the bad state. Action B pays 4 in the good state and 3 in the bad state. The good state has prior probability 2 5 . What is the expected value of perfect information about the state before acting?概率中等derivation未尝试面试订阅5916Most You Would Pay for a Perfect TestA product launch pays +30 if the market is receptive and -12 if it is not; receptivity has prior probability 3 10 . You may instead shelve the product for 0. A consultant offers a perfectly accurate test that reveals the true market state before you decide. What is the most you should be willing to pay for this test?概率中等derivation未尝试免费5917Free Peek Before Calling the Bigger BoxTwo boxes each independently contain an amount drawn uniformly from \ 1,2,3,4\ . You must guess which box holds the strictly larger amount; a correct guess pays 1 and a tie or wrong guess pays 0. Before guessing you may take a free peek at the contents of one box (your choice of which). By how much does this peek increase your probability of a correct guess compared with guessing blind?概率简单derivation未尝试免费5918Defective-Batch Inspection With an Imperfect DetectorA batch is defective with prior probability \frac14. Accepting a good batch pays +20; accepting a defective batch pays -40; rejecting pays 0. Before deciding you may run a detector that flags 'defective.' It flags a truly defective batch with probability 9 10 and a good batch with probability \frac15 (false positive). What is the value of running the detector (the increase in expected payoff from using it optimally)?概率困难derivation未尝试面试订阅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