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Fair value after an informed buy

MediumConditional Probability & ExpectationMarket Making & Betting Games~5m

A stock is worth either $100 or $110, each equally likely a priori. Of the order flow, 40%40\% is informed (they buy if the true value is $110, sell if it is $100); the other 60%60\% is uninformed and buys or sells with equal probability. A single buy order arrives. Conditioned on that buy, what is the fair value of the stock now (in dollars)?

Show hints (2)+
  1. A buy is evidence - update the fair value with Bayes' rule.
  2. Find P(buy∣V=110)P(\text{buy}\mid V{=}110) and P(buy∣V=100)P(\text{buy}\mid V{=}100) from the informed/uninformed mix, then take E[V∣buy]E[V\mid\text{buy}].

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