(1) Prove that the probability that exactly one of the events A or B occurs is equal to Pr[A] + Pr[B] 2 Pr[A B].

(2) A pair of n-sided fair dice are rolled (there is equal probability of rolling each number from {1, 2, . . . , n}).

What is the probability that the second dice lands on a higher value than does the first? [10 marks]

Given that the first dice is between 1 and m (for m n), what is the probability that the second dice lands on a higher value than does the first? [10 marks]

(3) Suppose that an insurance company classifies people into one of three classes - good risks, average risks, and bad risks. Their records indicate that the probabilities that good, average, and bad risk persons will be involved in an accident over a 1-year span are, respectively, 0.05, 0.15, and 0.30. If 20% percent of the population are good risk, 50% percent are average risk and 30% percent are bad risk,

What is the probability that a person has an accident in a fixed year? [10 marks]

If a policy holder had no accidents in the 1-year span, what is the probability that they are good risk? [5 marks]

(4) Two integers are selected at random from {1, 2, . . . , n}.

What is the probability that the integers are consecutive? [5 marks]

What is the probability that their sum is odd? [5 marks]