For a finite probability space 
let
This captures the intuition that the probability of an event is proportional to the number of ways that the event might occur.
The uniform assignment of probability involves counting. For small sample spaces this can be accomplished by examining all cases. Moderate size sample spaces can be inspected by a computer program.
Example: Suppose three six-face dice are tossed. What is the probability that three distinct numbers were selected?
Possible choices are (1,1,1) (1,1,2) (1,1,3) ... for a total of 216
cases.
There are 120 choices corresponding to the event that all numbers are
distinct (6 choices for the first number times 5 choices for the second
times 4 choices for the last one). Therefore the probability is 5/9.
The following BASIC program inspects all outcomes when five dice
are rolled, and counts how many are ``four of a kind".
Here is a portion of its output:
The program is written explicitly for tossing five dice, and you may want to modify it to answer similar question for any number of dice, and an arbitrary k-of-a-kind question.
As powerfull as the computers are, tossing five dies is not far from their current limitations! Typicall Pascal or C-realization of the above program runs about 15 times faster than the compiled QuickBasic code. Typical JavaScript version (executed on a Netscape) is about 12 times slower than the QuickBasic code. Imagine how long it would take to answer the same question when tossing 15 dies. (Whatever computer and programming language you use, in XX century it will take years) Exercises show the power of old-fashioned pencil-and-paper calculation based on counting techniques.
Counting techniques for arbitrary large probability spaces is the domain of combinatorics. The usual elementary ingredients are product rule, combinations, permutations; less elementary methods use generating functions, combinatorial identities, etc. Short review in Section Elementary combinatorics recalls the most elementary counting techniques.
The following exercises test your counting skills. You should not continue
untill you get the right answer.
Exercises
