Answer all questions. To maximize your chance to get partial credit, be sure to explain the reasoning behind your answers.

1) [20 points total] Two fair 6-sided dice, each with faces numbered 1 to 6, are rolled. What is the probability that

a) the product of the two faces is equal to 8? [10 pts]

b) the sum of the two faces is divisible by 4? [10 pts]

2) [20 points total] In straight poker, one is dealt a hand of 5 cards from an ordinary 52 card deck (no jokers). The order in which these 5 cards are dealt is irrelevant for determining the value of the hand.

a) What is the total number of distinct hands that can possibly be dealt? [7 pts]

b) How many of these possible hands will have all 5 cards in the same suit? Such a hand is called a flush. [6 pts]

c) What is the probability to be dealt a flush? [7 pts]

3) [30 points total] Assume that in a certain country it is found that the probability to give birth to a boy baby is 7/12, while the probability to give birth to a girl baby is 5/12.

a) What is the probability that a family will have their first two children boys, and the third child a girl? [5 pts]

b) What is the probability that a family of three children will have exactly two boys and one girl? [8 pts]

c) What is the probability that a family of five children will have exactly three boys and two girls? [10 pts]

d) What is the probability that in a family of four children, at least one will be a girl? [7 pts]

4) [30 points total] Assume that in a certain city, the distribution of families owning pets is as follows: 15% own no pet, 25% own exactly one dog and no cats, 15% own exactly two dogs and no cats, 20% own exactly one cat and no dogs, 10% own exactly two cats and no dogs, and 15% own exactly one dog and one cat. Assume that there are no other combinations.

a) What is the average number of dogs owned per family? [5 pts]

b) What is the standard deviation of the number of dogs owned per family? [5 pts]

c) What is the average number of cats owned per family? [5 pts]

d) In a group of 10,000 families are selected at random, what is the likely range for the total number of dogs that one would find? [8 pts]

e) In this group of 10,000 families, what is the likely range for the total number of families that own no pet at all? [7 pts]