Combinatorics and Probability
Section 3.4


Using rules of probability cuts work by not having to count the outcomes and sample space.  Using combinatorics from Chapter 2 is another alternative.


Recall Combinatorics are the Fundamenatal Counting Principle (FCP), permutations and combinations.


FCP: Multiply each category of choices by the number of choices.


Permutations:  Selecting more than one item without replacement where order is important.


Combinations: Selecting more than one item without replacement where order is not important.


Example 1: 

A lottery has 53 numbers from which seven are selected.  What is the probability of picking all seven numbers?

Find n(S).  Counting is tedious and impractical, use combinatorics.  Selecting more than one item from a group without replacement and order does not matter. 

        n C r =  53 C 7 =  53!             =  154,143,080 = n(S)

        n(picking 7 out of 7) = 1


        P(picking 7 out of 7) =  1/154,143,080


Example 2: 

Find the probability of being dealt three kings in a five-card hand in a 52-card standard deck.  (No jokers)

Sample space: how many 5-card hands are there?


        n(S) = 52 C 5 = 2,598,960

Find n(3 kings):          there are two categories, kings and non-kings.

          4 C 3  x   48 C 2 = 4 x 1128 = 4512


Notice the sum of the numbers in front of C totals 52 and the number after totals 5.  This is one form of bookkeeping and checking your work. n = 52, r = 5


        P(3 kings) = 4 C 3 x   48 C 2   =  4512          =  .001736     
52 C 5             2,598,960



Find the probability of dealing any 3 of a kind.

Since the possibility of 3 kings is .001736, then extend that to any 3 of a kind by multiplying by the number of different possibilities: 13

        4512 x 13 
                           = .02256





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