1.  The following table classifies 1456 people by their  sex and by whether or not   they favour a gun law. Compute the probabilities if one of these 1456 persons is selected randomly:

a.   ���(���1), b.  ���(���1|���1), c. ���(���1|���2),

d. Interpret your answers to parts b. and c.

[1041/1456, 392/633, 649/823]

2. Suppose that ���(���) = 0.7, ���(���) = 0.5, ���([��� ∪ ���]′) = 0.1. a. Find ���(��� ∩ ���).

b.   Give ���(���|���).

c.    Give ���(���|���).

[3/10, 3/5, 3/7]

3. Suppose that the genes for eye colour for a certain male fruit fly are (���, ���) and the genes for eye colour for the mating female fruit fly are (���, ���), where ���  and

��� represent red and white, respectively. Their offspring receive one gene for eye colour from each parent.

a. Define the sample space for the genes for eye colour for the offspring.

b. Assume that each of the four possible outcomes has equal probability. If an offspring ends up with either two red genes or one red and one white gene for eye colour, its eye will look red. Given that an offspring’s eyes look red, what is the conditional probability that it has two red genes for eye colour?

[1/3]

4. A small grocery store has 10 cartons of milk, 2 of which were sour. If you are going to buy the sixth carton of milk sold that day at random, compute the probability of selecting a carton of sour milk.

[1/5]

5. A drawer contains four black, six brown and eight olive socks. Two socks are selected at random from the drawer.

a. Compute the probability that both socks are the same colour.

b. Compute the probability that both socks are olive if it is known that they are the same colour.

[49/153, 4/7]

6. An urn contains four balls numbered 1 through 4. The balls are selected one at  a time without replacement. A match occurs if ball numbered ��� is the ���th ball selected. Let event ������ denote a match on the ���th draw, ��� = 1,2,3,4. Show that

a. ���(��� ) = 3!.

���

b.   ���(���

4!

∩ ��� ) = 2!.

��� ��� 4!

c.  ���(������

∩ ������

∩ ������

) = 1!.

4!

d. The probability of at least one match is

1 1 1

���(���1 ∪ ���2 ∪ ���3 ∪ ���4) = 1 − 2! + 3! − 4! .

e. Extend this exercise so that there are ��� balls in the urn. Show that the

probability of at least one match is

1 1 1

���(���1 ∪ ���2 ∪ ⋯ ∪ ������) = 1 − 2! + 3! − 4! + ⋯ +

(−1)^���+1

���!

1

= 1 − (1 − +

1!

1 1

−

2! 3!

+ ⋯ +

(−1)^���

) .

���!

f. What is the limit of this probability as ��� increases without bound?

[1 − 1/���]

7. Let ��� and ��� be independent events with ���(���) = 1/4 and ���(���) = 2/3. Compute: (a) ���(��� ∩ ���), (b) ���(��� ∩ ���^′), (c) ���(���^′ ∩ ���^′), (d) ���[(��� ∪ ���)^′] and (e)

���(���^′ ∩ ���).

[1/6, 1/12, ¼, ¼, ½ ]

8. Die A has orange on one face and blue on five faces, Die B has orange on two faces and blue on four faces, Die C has orange on three faces and blue on three faces. These are unbiased dice. If the three dice are rolled, find the probability that exactly two of the three dice come up orange.

[2/9]

9. Each of the 12 students in a class is given a fair 12-sided die. In addition, each student is numbered from 1 to 12.

a. If the students roll their dice, what is the probability that there is at least one “match” like student 4 rolls a 4?

b. If you are a member of this class, what is the probability that at least one of the other 11 students rolls the same number as you do?

11 12

1 − ( )

12

11 11

, 1 − (   ) ]

12

10. Bowl ���1 contains 2 white chips, bowl ���2 contains 2 red chips, bowl ���3 contains 2 white and 2 red chips, and bowl ���4 contains 3 white chips and 1 red chip. The probabilities of selecting bowl ���1, ���2, ���3 or ���4 are ½, ¼, 1/8 and 1/8, respectively. A bowl is selected using these probabilities, and a chip is then drawn at random. Find

a. ���(���), the probability of drawing a white chip.

b. ���(���1|���), the conditional probability that bowl ���1 had been selected, given that a white chip was drawn.

[21/32, 16/21]

11. At a hospital’s emergency room, patients are classified and 20% of them are critical, 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 1% die. Given that a patient dies, what  is the conditional probability that the patient was classified as critical?

[60/95]