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Suppose a box contains 20 coins- 14 of them are fair i.e. both Head (H) and Tail (T) are equally likely, 3 of them have both sides H, and 3 of them have both sides T. A coin is picked randomly and tossed- if its H, you get Rs. 1700; else you get 1200. You are not allowed to inspect the coin, you can just see the outcome of tosses. Further suppose first toss happens to be T. Now what is the maximum amount that you will be ready to pay to play this game for the second toss (same coin)? Assume that you are a expected utility maximiser with a risk averse attitude.

Question

Suppose a box contains 20 coins- 14 of them are fair i.e. both Head (H) and Tail (T) are equally likely, 3 of them have both sides H, and 3 of them have both sides T. A coin is picked randomly and tossed- if its H, you get Rs. 1700; else you get 1200. You are not allowed to inspect the coin, you can just see the outcome of tosses. Further suppose first toss happens to be T. Now what is the maximum amount that you will be ready to pay to play this game for the second toss (same coin)? Assume that you are a expected utility maximiser with a risk averse attitude.

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Solution

To solve this problem, we first need to calculate the probabilities of picking each type of coin and getting a certain outcome.

  1. Probability of picking a fair coin (P(F)) = 14/20 = 0.7
  2. Probability of picking a coin with both sides H (P(HH)) = 3/20 = 0.15
  3. Probability of picking a coin with both sides T (P(TT)) = 3/20 = 0.15

Given that the first toss is T, we can update our probabilities. The coin cannot be a HH coin, so we only consider the fair coins and TT coins.

  1. Probability that the coin is fair given that we got T (P(F|T)) = P(T|F)P(F) / P(T) Here, P(T|F) is the probability of getting T if the coin is fair, which is 0.5. P(T) is the total probability of getting T, which is the sum of the probabilities of getting T with a fair coin and a TT coin, i.e., 0.5P(F) + P(TT) = 0.50.7 + 0.15 = 0.5. So, P(F|T) = 0.50.7 / 0.5 = 0.7.

  2. Probability that the coin is TT given that we got T (P(TT|T)) = P(T|TT)P(TT) / P(T) Here, P(T|TT) is the probability of getting T if the coin is TT, which is 1. So, P(TT|T) = 10.15 / 0.5 = 0.3.

Now, we calculate the expected payoff for the second toss. If it's H, you get Rs. 1700; else you get Rs. 1200.

  1. Expected payoff if the coin is fair (E[F]) = 0.51700 + 0.51200 = Rs. 1450
  2. Expected payoff if the coin is TT (E[TT]) = 1*1200 = Rs. 1200

Finally, we calculate the overall expected payoff considering the updated probabilities of having each type of coin:

Expected payoff = P(F|T)E[F] + P(TT|T)E[TT] = 0.71450 + 0.31200 = Rs. 1375

So, if you are a risk averse expected utility maximiser, the maximum amount you should be willing to pay to play this game for the second toss is Rs. 1375.

This problem has been solved

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