STAT 220: Basic Statistics for Quantitative Students

1. Do exercises 7.18, 7.36, 7.49, 8.60, and 8.63.

2. Recall that there are 52 cards in a standard deck. Each card has one of 13 values (2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, or A) and one of 4 suits (hearts, spades, diamonds, clubs).

   There are C(52,5)=2,598,960 different 5-card poker hands possible from a 52-card deck. Calculate (explain reasoning and give a number) how many 5-card poker hands result in:

   • A full house (three-of-a-kind of one value plus a pair of a different value)
   • Three-of-a-kind (where the other two cards are distinct from each other)
   • A flush that is not a straight flush (all five cards have the same suit, excluding any flush in which the five values are consecutive, such as A, 2, 3, 4, 5 or 10, J, Q, K, A. Note that an A could be used either before a 2 or after a K, so there are 10 possible sequences of values that would be considered a "straight")
   • Not required: If anyone wants to count the numbers of other kinds of poker hands, go right ahead!

3. A bridge hand consists of 13 cards from a 52-card deck. If you are dealt a bridge hand from a well-shuffled deck, what is the exact probability that your entire hand is of the same suit? Give your answer as a fraction in lowest terms, then as a decimal.

4. This problem is optional. Try it if you're up for a challenge.

   The game of craps is played as follows: The player rolls two dice.

   • If the sum is 7 or 11, the game is over and the player wins.
   • If the sum is 2, 3, or 12, the game is over and the player loses.
   • If the sum is 4, 5, 6, 8, 9, or 10, then this sum becomes the player's "point" and the game continues as follows:
     • The player continues to roll the two dice until either a sum of either 7 or the "point" comes up.
     • If the "point" comes up, the game is over and the player wins.
     • If 7 comes up, the game is over and the player loses.
     • If anything other than 7 or the "point" comes up, nothing happens and the player keeps rolling.
   What is the probability of winning a game of craps? Give your answer as a fraction in lowest terms and as a decimal.