We have three boxes, each of which contains 5 balls. The balls are either red or black. The amount of the red balls are respectively 2, 3 and 4 in the first, second and the third box. We flip a fair coin and if it comes heads, we select either the first or the second box with equal chances. If it is tails, then we select either the first or the third box with equal chances. Finally, we select a ball from the chosen box.

(a) What is the probability of choosing the first box?

(b) What is the probability of having a red ball in the end of this procedure?

(c) What is the probability of having a red ball given that the coin is heads?

(d) What is the probability of having a red ball given that the coin is tails?

(e) If a red ball is chosen in the end of this procedure, what is the probability that the coin outcome was heads?