please the following questiond carefully with detailed descriptions:

1. 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] = 1/6

Pr[Green has the opportunity] = 2/6

Pr[Scarlet has the opportunity] = 3/6

Exactly one has the revolver with the following probabilities, regardless of who has an opportunity: Pr[Plum has the revolver] = 4/8

Pr[Green has the revolver] = 3/8

Pr[Scarlet has the revolver] = 1/8

Draw a tree diagram for this problem. Indicate edge and outcome probabilities. [15 marks]

What is the probability that Colonel Mustard is shot? [5 marks]

What is the probability that Colonel Mustard is shot, given that Miss Scarlet does not have the revolver? [5 marks]

What is the probability that Mr. Green had an opportunity, given that Colonel Mustard was shot? [5 marks]

2.Prove the following statements.

For an event A and two mutually exclusive events E1 and E2, prove the following statement.

Pr[E1 E2|A] = Pr[E1|A] + Pr[E2|A]

implying that the union-rule for mutually exclusive events also works if we condition on the event A.

For an event A and two mutually exclusive events E1 and E2, prove the following statement.

Pr[A|E1 E2] = (Pr[A|E1] Pr[E1] + Pr[A|E2] Pr[E2]) /(Pr[E1] + Pr[E2])

Prove that the above rule generalizes the law of total probability, i.e. if E1E2 = S and E1E2 = ,

then the above equation recovers the law of total probability, i.e.

Pr[A] = Pr[A|E1] Pr[E1] + Pr[A|E1c] Pr[E1c ], where E1 and E1c are complement events.

3.A class consisting of 4 men and 12 women is randomly divided into four groups (labelled Group A, Group B, Group C and Group D) each of size 4.

What is the number of possible groups that can be formed? [5 marks]

Suppose we impose the restriction that each group needs to have exactly 1 man, what is the number of possible groups that can be formed? [10 marks]

Use the above results to calculate the probability that each group includes exactly man?