Part 1:
1. Find the sample space for a committee of two chosen from Alice (A), Bill (B), Carol (C), and David (D).
{(A, B), (A, C), (A, D), (B, C), (B, D), (C, D)}
{(A, B), (C, D)}
{(A, B), (A, D), (B, C), (C, D)}
{A, B, C, D}
Find the sample space if both sexes must be represented.
{(A, B), (A, C), (A, D), (B, C), (B, D), (C, D)}
{(A, B), (A, D), (B, C), (C, D)}
{(A, B), (C, D)}
{A, B, C, D}
2. Two dice are rolled. E is the event that the sum is even, F is the event of rolling at least one six, and G is the event that the sum is eight, List the outcomes for the following events:
(a) E ∩ F
Φ
{(2, 6), (4, 6), (6, 6))
{(6, 2), (6, 4), (6, 6), (2, 6), (4, 6)}
{(2, 2), (4, 4), (6, 6)}
(b) Ec ∩ G
{(6, 2), (6, 4), (6, 6), (2, 6), (4, 6)}
{(2, 2), (4, 4), (6, 5)}
Φ
{(2, 6), (4, 5), (6, 6)}
3. If a committee of 3 is to be chosen at random from a class of 11 students, what is the probability of any particular committee being selected?
What if the committee is to consist of a president, a vice president, and a treasurer? (Enter your answer as a fraction.)
4. In a town, 36% of the citizens contributed to the Republicans, 42% contributed to the Democrats, and 15% contributed to both. What percentage contributed to neither party?
5. A box contains 6 red and 7 green marbles. You reach in and remove 3 marbles all at once.
(a) Find the probability that these 3 marbles are all red. (Enter your answer as a fraction.)
(b) Find the probability that these 3 marbles are all of the same color. (Enter your answer as a fraction.)
6. An elevator has 6 people and makes 9 stops. What is the probability that no two people get off on the same floor? (Enter your answer as a fraction.)
7. The U.S. Senate consists of 100 members, 2 from each state. A committee of 8 senators is formed. What is the probability that it contains at least one senator from your state?
8. Use the given values to find the following. (Enter your answers as fractions.)
P(A) = 0.5, P(B) = 0.5, P(A ∩ B) = 0.1
(a) P (A given B)
(b) P (B given A)
9. Use the given value to find the following. (Enter your answers as fractions.)
P(A) = 0.4, P(8) = 0.5, P(A ∪ B) = 0.7
(a) P (A given 8)
(b) P (B given A)
10. A box contains 4 white, 2 red, and 5 black marbles. One marble is chosen at random, and it is not black. Find the probability that it is white. (Enter your answer as a fraction.)
11. Suppose that 60% of drivers are "careful" and 40% are "reckless." Suppose further that a careful driver has a 0.3 probability of being in an accident in a given year, while for a reckless driver the probability is 0.5. What is the probability that a randomly selected driver will have an accident within a year? (Enter your answer to two decimal places.)
12. Two students are registered for the same class and attend independently of each other, student A 90% of the time and student B 70% of the time. The teacher remembers that on a given day at least one of them is in class. What is the probability that student A was in class that day? (Round your answer to three decimal places.)
13. For the experiment of tossing a coin twice, find whether events
A: Heads on the first toss
B: Different results on the two tosses are independent or dependent.
Independent
Dependent
14. Four dice are rolled.
(a) Find the probability of getting all sixes. (Enter your answer as a fraction.)
(b) Find the probability of getting all the same outcomes. (Enter your answer as a fraction,)
(c) Find the probability of getting all different outcomes. (Enter your answer as a fraction.)
Part 2:
1. A box contains three marbles, one red (R), one green (G), and one blue (B). A first marble is chosen, its color is recorded, and then it is replaced in the box and a second marble is chosen, and its color is recorded. Find the sample space.
{(R, R), (R, G), (R, B), (G, R), (G, G), (G, 13), (B, R), (B, G), (B, B)}
{R, G, B}
{(R, G), (R, B), (G, R), (G, B), (B, R), (B, G)}
{(R, B), (B, G), (G, R)}
Find the sample space if the first marble is not replaced before the second is chosen.
{(R, R), (R, G), (R, B), (G, R), (G, G), (G, B), (B, R), (B, G), (B, B))
{R, G, B}
{(R, G), (R, B), (G, R), (G, B), (B, R), (B, G)}
{(R, B), (B, G), (G, R)}
2. If a committee of 3 is to be chosen at random from a class of 14 students, what is the probability of any particular committee being selected? (Enter your answer as a fraction.)
What if the committee is to consist of a president, a vice president, and a treasurer? (Enter your answer as a fraction.)
3. In a town, 33% of the citizens contributed to the Republicans, 44% contributed to the Democrats, and 11% contributed to both. What percentage contributed to neither party?
4. A college survey claimed that 62% of students took English composition, 47% took calculus, 15% took both, and 9% took neither. Show that these figures cannot be correct.
The real percentage that took neither class is 6, not 91Y0 as claimed by the survey.
The real percentage that took both classes is 6, not 15% as claimed by the survey.
The real percentage that took both classes is 9, not 15% as claimed by the survey.
The real percentage that took neither class is 15, not 9% as claimed by the survey.
5. A box contains 5 red and 6 green marbles. You reach in and remove 3 marbles all at once.
(a) Find the probability that these 3 marbles are all red. (Enter your answer as a fraction.)
(b) Find the probability that these 3 marbles are all of the same color. (Enter your answer as a fraction.)
6. An elevator has 5 people and makes 9 stops. What is the probability that no two people get off on the same floor? (Enter your answer as a fraction.)
7. The U.S. Senate consists of 100 members, 2 from each state. A committee of 6 senators is formed. What is the probability that it contains at least one senator from your state? (Round your answer to two decimal places.)
8. Use the given values to find the following. (Enter your answers as fractions.)
P (A) = 0.7, P (B) = 0.3, P(A ∩ B) = 0.2
(a) P (A given B)
(b) P (B given A)
9. A box contains 3 white, 2 red, and 5 black marbles. One marble is chosen at random, and it is not black. Find the probability that it is white. (Enter your answer as a fraction.)
10. You will take either a basket-weaving course or a philosophy course, depending on what your advisor decides. You estimate that the probability of getting an A in basket weaving is 0.80, while in philosophy it is 0.70. However, the chances of your advisor choosing the basket-weaving course is only 30%, while there is an 70% chance that he will put you in the philosophy course. What is the probability that you end up with an A? (Enter your answer to three decimal places.)
11. Suppose that 80% of drivers are "careful" and 20% are "reckless." Suppose further that a careful driver has a 0.1 probability of being in an accident in a given year, while for a reckless driver the probability is 0.5. What is the probability that a randomly selected driver will have an accident within a year? (Enter your answer to two decimal places.)
12. Three dice are rolled.
(a) Find the probability of getting all sixes. (Enter your answer as a fraction.)
(b) Find the probability of getting all the same outcomes. (Enter your answer as a fraction.)
(c) Find the probability of getting all different outcomes. (Enter your answer as a fraction.)