A coin is biased so that the chance of a head being tossed is three times that of a tail occurring. In a game the coin is tossed three times.
(a) Show the outcomes using a tree diagram and write the probabilities along the branches.

Step 1 ：Let the probability of getting a tail be x, then the probability of getting a head is 3x. Since the total probability of all possible outcomes in a single toss must equal 1, we can write the equation: \(3x + x = 1\)

Step 2 ：Solve for x: \(x = 0.25\)

Step 3 ：Find the probabilities of getting a head and a tail in a single toss: \(P(H) = 0.75\) and \(P(T) = 0.25\)

Step 4 ：Create a tree diagram for three tosses and calculate the probabilities along the branches:

Step 5 ：\(P(HHH) = 0.75 \times 0.75 \times 0.75 = 0.421875\)

Step 6 ：\(P(HHT) = 0.75 \times 0.75 \times 0.25 = 0.140625\)

Step 7 ：\(P(HTH) = 0.75 \times 0.25 \times 0.75 = 0.140625\)

Step 8 ：\(P(HTT) = 0.75 \times 0.25 \times 0.25 = 0.046875\)

Step 9 ：\(P(THH) = 0.25 \times 0.75 \times 0.75 = 0.140625\)

Step 10 ：\(P(THT) = 0.25 \times 0.75 \times 0.25 = 0.046875\)

Step 11 ：\(P(TTH) = 0.25 \times 0.25 \times 0.75 = 0.046875\)

Step 12 ：\(P(TTT) = 0.25 \times 0.25 \times 0.25 = 0.015625\)

Step 13 ：\(\boxed{P(HHH) = 0.421875, P(HHT) = 0.140625, P(HTH) = 0.140625, P(HTT) = 0.046875, P(THH) = 0.140625, P(THT) = 0.046875, P(TTH) = 0.046875, P(TTT) = 0.015625}\)