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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