Math 207 – Counting-Method Problems

Counting methods can sometimes be useful in calculating probabilities (especially when the outcomes are equally likely, yet the sample space is too large to list). It’s typically easiest to first count the total number of possible outcomes for the experiment (this is the number you place in the denominator of the fraction). When calculating the number of possible outcomes in your specific event of interest (the number for the numerator), 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.) 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).
    1. How many possible poker hands are there? (Note: Are poker hands considered “ordered”?)

 

    1. What is the probability of a full house (i.e., three cards of one rank and two cards of another rank)?

 

    1. What is the probability of a straight (i.e., five cards in order—the ace can be used as a high card or a low card, but a straight can’t be, say Queen, King, Ace, Two, Three), but not a straight flush (i.e., five cards in order and all of the same suit)? Hint: How many starting-spots are there for the straight?

 

    1. What is the probability of 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.

 

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

                                                               i.     

 

                                                            ii.     

 

  1. Suppose you have 3 distinct Green Bay Packers fans and 3 distinct Chicago Bears fans.
    1. In how many ways can the fans sit in a row? (Note: Now order matters.)

 

    1. What is the probability of the fans sitting in a row such that the Packers fans are seated together and the Bears fans are each seated together?

 

    1. What is the probability of the fans sitting in a row such that the Packers fans are seated together?

 

    1. What is the probability of the fans sitting in a row such that no two fans of the same team are seated together?

 

  1. A person has 8 friends, of whom only 5 will be (randomly) invited to a party (perhaps he only has 6 place settings).
    1. How many choices are there if two of the friends are fighting and won’t attend together? (Here you are simply counting, not determining a probability.) Hint: Can you break down this event into mutually exclusive events (then you can add the number of possibilities 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.     

 

    1. How many choices if two of the friends will only attend together (perhaps they just started dating and are currently inseparable)?