# Calculating Probability

Let's calculate the probability of rolling two sixes.  We can start by looking at all the pairs that can possibly be rolled:

 1-1 1-2 1-3 1-4 1-5 1-6 2-1 2-2 2-3 2-4 2-5 2-6 3-1 3-2 3-3 3-4 3-5 3-6 4-1 4-2 4-3 4-4 4-5 4-6 5-1 5-2 5-3 5-4 5-5 5-6 6-1 6-2 6-3 6-4 6-5 6-6

There are thirty-six possibilities all together (notice that 36=6*6=6^2).  Out of those, there is only one pair that is 6-6, so the odds of rolling 6-6 is 1/36.

Now consider the probability of rolling a number less than three followed by a number greater than two.  How many ways can we do this?  For the first number, we can roll a one or a two.  For each of these two choices, we can roll either a 3, 4, 5, or 6:

 1-3 1-4 1-5 1-6 2-3 2-4 2-5 2-6

So there are eight ways that we can get the desired events (two ways to get a number less than three followed by four ways to get a number greater than two--2*4=8).  Since we have already shown that there are 36 possible combinations, the probability of our two independent events is 8/36 = 2/9.