Answer all questions. To maximize your chance to receive 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 the product of the two faces that land on top is:

a) exactly equal to 12? [10 pts]

b) greater than 16? [10 pts]

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2) [40 points total]

On a casino roulette wheel, there are the numbers 1 to 36, plus the numbers 0 and 00. The numbers 1 to 36 are colored half red and half black (0 and 00 are green). Assume that on each spin of the wheel, any number is equally likely to come up.

a) A gambler places his bet that a black number will come up on the next spin. What is the probability that he will win his bet? [10 pts]

b) What is the probability that the first 3 spins of the wheel all come up black? [10 pts]

c) What is the probability that in 3 spins of the wheel at least one black comes up? [10 pts]

d) What is the probability that in 6 spins of the wheel, exactly 3 come up black? [10 pts]

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3) [40 points total]

In a certain city it is found that the fraction of families that own no cars is 0.12; the fraction that own one car is 0.42; the fraction that own two cars is 0.38; and the fraction that own three cars is 0.08.

a) What is the mean number of cars owned per family? [10 pts]

b) What is the standard deviation of cars owned per family? [10 pts]

c) If a group of 10,000 families is selected at random, what is a likely range for the number of cars that one would find? [10 pts]

d) In this group of 10,000 families, what is a likely range for the number of families that own no cars? [10 pts]