Please answer these four questions step by step, and some explanation

(1) [20 marks] Three cards are drawn one after the other from an ordinary 52card deck without replacement (once a card is drawn, it is not placed back in the deck). Compute the probability that (i) All of the three cards is a heart. [5 marks] (ii) None of the three cards is a heart. [5 marks] (iii) Exactly one of the three cards is a heart. [10 marks] (2) [10 marks] Suppose we have is: computer servers and m jobs that needs to be assigned to these servers. We are using a random allocation strategy such that each job is randomly assigned to a server. A server fails if there is more than one job assigned to it. Calculate the probability that no server fails. (3) [25 marks] A class consisting of 4 graduate and 12 undergraduate students is randomly divided into four groups of 4. o What is the number of possible groups that can be formed? [10 marks] 0 Suppose we impose the restriction that each group needs to have exactly one graduate student, what is the number of possible groups that can be formed? [10 marks] 0 What is the probability that each group includes exactly one graduate student? [5 marks] (4) [20 marks] Let us consider a game played between two players rolling a \"standard\" dice. The players roll the dice alternatively, and the rst player that rolls a 6 wins the game. I What is the probability that Player 1 (who starts the game) wins with their rst roll of the dice? [2 marks] 0 What is the probability that Player 1 (who starts the game) wins with their second roll of the dice? [5 marks] 0 What is the probability that Player 1 (who starts the game) eventually wins the game? [13 marks]