Could you please answer these four questions(6,7,8,11) step by step and some explanation?

Using conditional/total probability and independence knowdge, each question is not long :)

(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] = GJICADCDI MQIH Pr[Scarlet has the opportunity] = 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] = Pr[Scarlet has the revolver] = OOIi-'CXJIDJQJIv-lh 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 affects 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 \"no\" 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 (has a smaller probability of making a misdiagnosis)? [1 marks] (8) [15 marks] The SFU football team has probability 0.3 of winning against Tier 1 teams, probability 0.4 of winning against Tier 2 teams and probability 0.5 of winning against Tier 3 teams. Half the teams in the league are Tier 1 while a quarter of them are Tier 2 and a quarter of them are Tier 3. The odds of an event A is given by: Pr[A] odds[A] = m o What are the odds that the SFU football team wins against a randomly chosen team? [10 marks] 0 Given SFU wins the game, what is the probability that the opponent was a Tier 2 team? [5 marks] (11) [15 marks] We roll a \"standard\" dice twice. 0 A1 is the event that the rst roll is 1 and BI is the event that the sum of the numbers on the two dice is equal to 5. Calculate PI[A1], Pr[Bl] and Pr[A1 Bl] and ascertain whether the events A1 and BI independent? [5 marks] 0 A2 is the event that the maximum of the two rolls is 2 and 32 is the event that minimum of the two rolls is 2. Calculate Pr[A2], Pr[Bg] and Pr[A2 n 32] and ascertain whether the events A2 and 32 independent? [10 marks]