# Bayes theorem coin problem

Suppose you are told that a given coin is biased 2/3 : 1/3 , but you don’t know which way: it might be biased 2/3-heads 1/3-tails, or it might be biased 1/3-heads 2/3-tails. Your a priori belief is that there is a 3/4 chance that the coin is heads-biased and a 1/4 chance that the coin is tails-biased. You plan to flip the coin four times and update your initial subjective belief based on Bayes Theorem. You then guess the bias of the coin (heads-biased or tails-biased) according to which of these two hypotheses has the larger final subjective probability. For each of the two possible situations (i.e., the coin is really heads-biased or it is really tails-biased), compute the probability that your guess will be wrong.