1. (3 points) Suppose two dice are rolled. What is the probability that the sum of the two dice is 4?
2. (4 points) Suppose two dice are rolled. What is the probability that the sum of the two dice is either a 3 or a 7?
3. (4 points) Suppose we flip 3 coins. What is the probability that exactly 1 coin lands heads up? It might help to write down the entire sample space in roster notation.
4. (4 points) How many integers are there from 1-1000, inclusive, that have no repeated digits?
5. A lunch special consists of a salad, a sandwich, and a drink. The choices for the salad are greek, traditional, and Caesar. The choices for the sandwich are ham, turkey, roast beef, or vegetarian. The choices for the drink are soda, tea, coffee, or juice.
(a) (3 points) How many different lunch specials are possible? Think: Tree diagram
(b) (3 points) If a lunch special is chosen at random, what is the probability that it includes a traditional salad.
6. Suppose a password to a computer site can consist of three, four, or five digits (0-9).
(a) (3 points) How many different passwords are possible if repetition of digits is permitted?
(b) (4 points) How many different passwords are possible if repetition of digits is not allowed?
(c) (4 points) What is the probability that a randomly chosen password will have at least one repeated digit?
7. (4 points) What is the probability that an integer from 1 to 1000, inclusive, chosen at random, is a multiple of 3 or a multiple of 7?
8. Eight people, including Joe and Ellen, are going to the movies and will sit in a row of 8 chairs.
(a) (3 points) In how many ways can the eight people seat themselves?
(b) (4 points) In how many ways can they seat themselves if Joe and Ellen must sit together? Note: The order here could be Joe Ellen or Ellen Joe.
(c) (2 points) If all eight people sit down randomly, what is the probability that Joe and Ellen are seated next to each other?
9. (3 points) If we choose a random integer from 1-10,000, inclusive, what is the probability that the integer contains at least one 6?
Hint: Think about what the complement of this event would be.
10. The following gives the results from a survey of the numbers of freshmen and sophomores at a certain community college:
| Males | Females | Total | |
| Freshmen | 2000 | 3000 | |
| Sophomores | 2500 | 2800 | |
| Total |
(a) (2 points) What is the probability that a randomly chosen student is a freshman and female?
(b) (3 points) What is the probability that a randomly chosen student is a freshman, given that she is a female?
(c) (4 points) What is the probability that a randomly chosen student is a male or a sophomore?
Hint: Are these events disjoint?
11. (4 points each) Suppose a jar contains 10 marbles: 5 green, 3 blue, and 2 yellow, and we are to randomly choose two marbles from the jar. Calculate the following probabilities.
(a) The probability that if two marbles are chosen WITH replacement, that both are blue:
(b) The probability that if two marbles are chosen WITHOUT replacement, that both are blue:
(c) The probability that if two marbles are chosen WITHOUT replacement, that at least one is blue:
12. BONUS (1 point): What are your plans for Thanksgiving?