Student A in the stochastics class is quizzing. The quiz is a question of 10, all three-way multiple-choice questions (multiple-choice questions with three choices). Independently in each problem, A knows the answer with a 50% probability or does not know the answer with a 50% probability. If you know the answer, the probability of matching the answer is 1, and if you do not know, the probability of matching is 1/3. If A has hit the first problem, then find the probability that A has known the answer to the problem.

Use the Bayesian rule, with A being the probability P(K), the probability P(K') that you didn't know, and the probability P(C) that you answered.