1. Find the number of ways to draw:
a. A 5-card hand consisting of spades only
b. A 4-card hand with 2 aces and two queens
2. There are 10 puppies in a cage, of which 6 are male and 4 are female. If you select 3 puppies at random, what is the probability that they will all be the same gender?
3. To play a lottery game a player must choose 6 numbers between 1 and 49. The order is not important. Six numbers between 1 and 49 will turn out to be winning numbers for a particular game. If you play this lottery game, what is the probability of
a. Choosing the 6 winning numbers and winning the lottery?
b. Matching exactly 4 of the winning numbers?
4. There are 25 people in a room. What is the probability that at least two have the same birthday?
5. Consider a game that consists of rolling a regular die. You pay $10 to play the game and the $10 is not returned. If you roll a 1 or 2, you win nothing. If you roll a 3, 4, or 5, you win $8; if you roll a 6 you win $20. What is your expected value for this game?
6. The game of 5-spot Keno is similar to 4-spot Keno. 80 numbers are displayed on a board, and 20 of these numbers will be chosen to be the winning numbers. To play, the player selects 5 numbers on his or her playing card. Suppose a player pay $1 to play, and the $1 is not returned. The payoffs made by the Dunes Casino in Las Vegas for each of the possible outcomes are listed here:
|
Outcome
|
Payoff
|
|
0, 1, or 2 winning numbers
|
$0
|
|
3 winning numbers
|
$2
|
|
4 winning numbers
|
$20
|
|
5 winning numbers
|
$480
|
a. What is the expected value of this game?
b. What is the house edge for this game, expressed as a percentage of $1?
7. A man has blood type B. The man's mother was type AB and his father was type A. If his wife has blood type O and is expecting a baby, what is the probability the baby will have blood type
a. A?
b. B?
c. O?
d. AB?
8. For some types of plants, when plants with different colored flowers are crossbred, one color does not dominate over the other. Instead, the offspring have flowers that are a blend of the colors of the 2 parent plants. Mendel found that when he crossed a snapdragon with red flowers with a snapdragon with white flowers, the resulting offspring had pink flowers. In other words, when a red snapdragon of type RR was crossbred with a white one of type WW, the resulting offspring of type RW had pink flowers. Suppose a white snapdragon is crossbred with a pink snapdragon. What is the probability that a resulting offspring will be
a. Red?
b. White?
c. Pink?
9. Determine whether the graph below has a Eulerian circuit, and if it does, give one. If it does not, explain why not. If the graph does not have a Eulerian circuit, determine whether it has a Eulerian path, and if it does, give one. If it does not, explain why not.
10. Suppose a garbage truck must pick up garbage in a neighborhood with streets laid as in the graph. The workers pick up from both sides of the street at the same time, so ideally they would like to travel down each street as few times as possible. Find a eulerization of the graph that will allow the garbage truck to make a reasonably efficient Eulerian circuit through the neighborhood.
11. 6 cards are dealt from a standard deck. What is the probability that the hand contains 3 cards of 1 rank, 2 cards of a 2nd rank, and 1 card of a 3rd rank?