Question 1 A carton of 12 rechargeable batteries contains two batteries that are defective.

(a) In how many ways can an inspector choose three (3) of the batteries?

(b) In how many ways can an inspector choose three (3) of the batteries and get none of the defective batteries?

(c) In how many ways can an inspector choose three (3) of the batteries and get one of the defective batteries?

(d) In how many ways can an inspector choose three (3) of the batteries and get both of the defective batteries?

Question2 A software engineer wants to move across the country from Vancouver to Montréal to work. On different days, she mails a cover letter and resume to three different companies in Montréal. Suppose a letter sent from Vancouver to Montréal has a probability of 0.75 of reaching Montréal within three (3) days.

(a) What is the probability that exactly two (2) of the three (3) letters will reach Montréal within three (3) days?
(b) What is the probability that at least one (1) of the three (3) letters will reach Montréal within three (3) days?
(c) If the three (3) letters are mailed together at the same time and location, how does your conclusion in part (a) change?

Question 3 Two software consulting firms V and W consider bidding on a large programming project, which may or may not be awarded depending on the amounts of the bids. Firm V submits a bid and the probability is 3/4 that it will get the project provided that firm W does not bid. The probability is 3/4 that W will bid, and if it does, the probability that V will get the project is 1/3.

(a) What is the probability that V will get the project?
(b) If V gets the project, what is the probability that W did not bid?

Question 4 Two balls, each equally likely to be coloured either red or blue, are put in an urn. At each stage one of the balls is randomly chosen, its colour is noted, and it is then returned to the urn. If the first two balls chosen are coloured red, what is the probability that
(a) both balls in the urn are coloured red;
(b) the next ball chosen will be red?

Question 5 Suppose that 5 independent trials, each of which results in any of the outcomes 0, 1, or 2, with respective probabilities 0.3, 0.5, and 0.2, are performed. Find the probability that both outcome 1 and outcome 2 occur at least once. (Hint: Consider the complementary probability.)