Can you help me to solve these three questions step by step? And some explanation?

(5) [25 marks] You shuffle a deck of cards and deal your friend 5 cards. 0 What is the probability that they have exactly one ace in their 5 cards? [5 marks] 0 What is the probability that they have the ace of spades in their 5 cards? [5 marks] 0 Suppose your friend says, \"I have the ace of spades\". What is the probability that they have exactly two aces? [5 marks] 0 Suppose your friend says, \"I have at least one ace\". What is the probability that they have at least two aces? [10 marks] (6) [30 marks] Professor Plum, Mr. Green, and Miss Scarlet are all plotting to shoot Colonel Mustard. If one of these three has both an opportunity and the revolver, then that person shoots Colonel Mustard. Otherwise, Colonel Mustard escapes. Exactly one of the three has an opportunity with the following probabilities: Pr[Plum has the opportunity] = Pr[Green has the opportunity] = Pr[Sca.rlet has the opportunity] = ONCOQINO'JI H Exactly one has the revolver with the following probabilities, regardless of who has an opportunity: Pr[Plum has the revolver] = Pr[Green has the revolver] = OOIHOOIOJOOI Lab Pr [Scarlet has the revolver] = 0 Draw a tree diagram for this problem. Indicate edge and outcome probabilities. [15 marks] 0 What is the probability that Colonel Mustard is shot? [5 marks] 0 What is the probability that Colonel Mustard is shot, given that Miss Scarlet does not have the revolver? [5 marks] 0 What is the probability that Mr. Green had an opportunity, given that Colonel Mustard was shot? [5 marks] (7) [15 marks] There is a rare disease called Nerditosis which aficts about 1 CS student in 1000. One symptom of Nerditosis is a compulsion to refer to everything using numbers or Greek symbols. Two professors claim that they can diagnose Nerditosis. o If the student has Nerditosis, Professor X can detect it with probability 0.99. o If the student does not Nerditosis, Professor X will erroneously detect it with probability 0.03. 0 Professor Y does not have the sophisticated equipment that Professor X has, but he knows that Nerditosis aects one in thousand students on average. Hence, when asked if a student has Nerditosis, Professor Y says \"yes\" with probability . 2 o What is the probability that Professor X misdiagnoses Nerditosis i.e. says \"yes\" when the student does not have it or ' 0\" when the student does indeed have it? [7 marks] 0 What is the probability that Professor Y misdiagnoses Nerditosis? [7 marks] 0 Which professor is more accurate in detecting Nerditosis? [1 marks]