Probability Theory—Chapter 1 Practice Problems

Notes: When calculating the number of possible outcomes, think carefully about how the event is defined. Does order matter or not matter? How can you break a more complex event into separate, simple pieces? (Then you can use the basic principle of counting to multiply together the separate possibilities to determine the overall number of possible outcomes in your event. Or, if you can break up the event into distinct pieces, you can add together the possibilities.) Be careful to rethink your answer to ensure you didn’t over- or undercount. Sometimes it can be helpful to write out a few possible outcomes in the event of interest, so you get a sense of an overall pattern and important things to think about. Lastly, there are often multiple correct methods to solve these problems (if you think of two methods, use them both, and then check if they agree—this is a good way to get a “reasonable check” on your answer).

 

1.       A poker hand is defined as five cards randomly drawn from a standard 52-card deck. Note that each 52-card deck has 13 ranks (2, 3, …, 10, jack, queen, king, ace) and each rank is represented in 4 suits (hearts, diamonds, clubs, and spades).

a.      How many possible poker hands are there?

 

b.      How many ways are there to get a “full house” (i.e., three cards of one rank and two cards of another rank)?

 

c.      How many ways are there to get a “three-of-a-kind” (i.e., exactly three cards of the same rank)? Note that a full house does not count as a three-of-a-kind.

 

d.      In regard to part c, explain why the following answers are incorrect (and how you could change them to make them correct):

                                                                           i.     

 

                                                                        ii.     

 

2.      (Problem 11 in the textbook) In how many ways can 3 (distinct) novels, 2 (distinct) mathematics books, and 1 chemistry book be arranged on a bookshelf if

a.      the books can be arranged in any order?

 

b.      the mathematics books must be together and the novels must be together?

 

c.      the novels must be together, but the other books can be arranged in any order?

 

3.      (Problem 20 in the textbook) A person has 8 friends, of whom 5 will be invited to a party.

a.      How many choices are there if 2 of the friends are feuding and will not attend together?

 

b.      In regard to part a, explain why the following answer is incorrect (and how you could change it to make it correct):

                                                                          i.     

c.      How many choices are there if 2 of the friends will only attend together?

 

4.      (Theoretical Exercise 12) Consider the following combinatorial identity:

a.      Present a combinatorial argument for this identity by considering a set of n people and determining, in two ways, the number of possible selections of a committee of any size and a chairperson for the committee. Hint:

                                                                          i.      How many possible selections are there of a committee of size k and its chairperson?

 

                                                                        ii.      How many possible selections are there of a chairperson and the other committee members?