1. To play a game: You have two bags. Then a fair coin is flipped. If the coin is tails, then a ball is
chosen at random from Bag 1. If the coin is heads, then a ball is chosen at random from Bag 2.
Let R, G and Y be the events that the chosen balls is red, green and yellow, respectively. Let T
be the event that the coin is tails.
1) Suppose we find the probabilities of choosing each colour are P(R) = 0.45, P(G) = 0.1 and
P(Y ) = 0.45. Determine the entropy of the probability distribution for the outcome of picking a
ball.
2) To win the game you need to pick a green ball and you have three tries, replacing the picked
ball after each attempt. As soon as you pick the green ball the game terminates and you win. Let
X be the random variable that represents the outcome of the game; possible values of X are G,
RG, and Y RR. Let Z be the number of attempts played which ranges from 1 to 3. Assuming that
that you replace the ball after each attempt, calculate H(X), H(Z), H(Z|X) and H(X|Z).