Compute two conditional probability scenarios

Solve both parts, include the formulas, and give a final numerical answer. A) A box holds 3 fair coins and 2 double-headed coins. You select one coin uniformly at random, toss it three times, and get H,H,H. What is $P(\text{coin is double-headed} \mid \text{3 heads observed})$? B) An urn has 5 red, 4 blue, and 3 green balls. You draw 3 balls without replacement. What is the probability that exactly two different colors appear and at least one of the balls is red? Overview: This question tests a candidate's grasp of conditional probability, Bayesian reasoning for posterior inference, and combinatorial counting in finite-sample probability problems.