Lottery games are random (by law), so they are obviously a good
example of random probability. Let's take an example lottery
where you must pick 6 unique numbers in any order from 1 to 49.
Remember that the first step is to count the number of possible
combinations: there are 49 choices for the first number. You
can't pick the same number twice, so there are only 48 choices
for the second number, 47 choices for the third number, and so on.
Therefore, the total number of possible combinations is
49*48*47*46*45*44 (or 49!/43! on your calculator), which
equals approximately 10.07 billion. However, you can pick
the 6 numbers in any order, and there are 720 possible ways
to arrange 6 numbers (6*5*4*3*2*1, or 6! on your calculator; now
you know what that x! button is for). Therefore, the overall
probability is 10.07 billion divided by 720 orders, or 13.98
million (probability math short-hand for this whole process is
"49 choose 6", or 49C6). Therefore, the odds of any given
set of numbers coming up in this type of lottery are roughly 1
in 14 million. And if you play two sets of numbers in
the same lottery draw, then your odds of winning are roughly 2 in
14 million, or 1 in 7 million.