Bayesian Decision theory

A person randomly chooses a coin with probability 0.5: Tossing coin 1 yields a head with a probability P(X1=H)=.3 (and tail with P(X1=T)=.7). Tossing coin 2 yields a head with a probability P(X2=H)=.6 (and tail with P(X2=T)=.4). You earn 1 dollar if you correctly guessed the coin and 0 dollars otherwise. Design the optimum decision rule and estimate your average earning.

Can anybody explain me ,how should i go about solving this problem?

• what is your experiment? is it one toss? or $n$ tosses, and you decide based on number of heads? Mar 25 '19 at 9:55
• Is this a homework question? If so, please add the self-study tag. Mar 25 '19 at 20:39
• Could the user share his findings on this question? I am new to Bayesian and trying to understand this problem as homework. It would be really helpful if the user shares the whole answer to this question. Mar 27 '20 at 19:48
• @user277892 - Can you let me know exactly which part you find unclear or need further help with? Then I can expand my answer to help you out. Mar 27 '20 at 20:10

Given that you know all the probabilities that govern your system, I don't think you need to apply anything "Bayesian" for this. You can simply calculate the probability of heads as

$$P(H) = P(H | X1) P(X1) + P(H | X2) P(X2)$$

You can use a similar calculation to calculate the probability of tails. Now that you have these probabilities, you can compute the expected earnings if you guess heads and if you guess tails, and base your decision rule on this calculation.

Please let me know if any of this is unclear.

Edit:

When you know $$P(H)$$ and $$P(T)$$ you can compute the expected earnings as $$E(\mathrm{earnings}) = P(H)*[\mathrm{earnings\ in\ case\ of\ }H] + P(T) *[\mathrm{earnings\ in\ case\ of\ }T]$$

Now the earnings in case of heads and tails depend on your decision rule, so you should compute the expected earnings for every possible decision you can make and then make the decision with the highest expected earnings.

• I calculated the probabilities but i'm not able to figure out how should i compute the expected earnings... Mar 25 '19 at 22:10
• I edited my answer so that you can hopefully solve your problem now. Mar 26 '19 at 6:53