1. You and your friends are playing a game where there are 10 cards:  5 are red, 3 are blue, and 2 are green.  Each player draws one card without replacement.  The player who draws a green card is the winner.

(a) If you have a choice of drawing first, second, or third, which position would you

choose? Justify your choice on the basis of probability.

(b) Now, suppose there are 4 green cards and 6 red cards in the deck. What position

would you now choose? Justify your choice on the basis of probability.

(c) Assuming that there is always at least one green card in the deck, what should be the number of green cards so that it would not matter for you to draw first or second? Justify your choice on the basis of probability.

2. You are the manager of a manufacturing plant and you are responsible for quality control.  Each day, a random sample of 10 items is selected from the production line and tested by a quality control technician. A item can be in good condition or defective, and the probability that an item is defective is 0.05.  You know by experience that the technician records the result of the test on an item correctly 80% of the time.

Let X be the number of items that were recorded as defective by the technician (in the sample of 10 tested items).

(a) Find the probability P(X = 1).

(b) If the technician recorded one defective item in the sample, what is the conditional

probability that there was no defective item in the sample?

3. Let P(A) = P(B) =  and P(A ∩ B) =  . Find the following:

(a) P(A\ ).

(b) P(A\ ∩ B\ ).

(c) P(B ∩ A\ ).

(d) P(A ∪ B\ ).

4.   (a) Consider a group of four students. Each student has a favourite colour, which can be any one of the colours of the rainbow (red, orange, yellow, green, blue, indigo,

violet), assuming that each colour is equally likely as a favourite colour.        What is the probability that no two students have the same favourite colour?

(b) Consider a group of seven students.  Each student has a favourite colour, which

can be any one of the seven colours of the rainbow (assuming again that each colour is equally likely as a favourite color).

i. What is the probability that no two students have the same favourite colour?

ii. What is the probability that some of them have the same favourite colour?

5. You have three bags of marbles.  Bag A contains 1 red and 4 blue marbles, Bag B contains 2 red and 3 blue marbles, and Bag C contains 3 red and 2 blue marbles.

(a) If one marble is drawn from each bag, find the probability that exactly two of the

three marbles are blue.

(b) If you randomly select one bag and draw one marble from it, find the probability

that the marble is blue.

6. A certain type of car model has a defective part in its engine, causing the engine to fail 10% of the time. However, the engine fails 2% of the time due to reasons other than the defective part. A dealership sells this car model and 0.5% of the cars sold have the defective part.

(a) What is the probability that a randomly selected car from the dealership will have

an engine failure?