Example: 7  
A pair of dice is rolled until a sum of either or is obtained. Find the probability that comes before .

Solution: 7  
A sum of can be obtained in ways, namely
from a total number of ways of throwing a pair of dice. If we let denote the event of obtaining a sum of , we have
Similarly, let the event be that of obtaining a sum of ; this can happen in ways, namely
so that
Finally, if we let be the event of obtaining neither a or a , we have
Now, we want a sum of to come before a sum of . Think about how this can happen. Every time you roll the pair of dice, you should either get a sum of or you should get neither a sum of nor .
Therefore, the following (mutually exclusive) sequences of throws leads us to a sum of before a sum of .
The required probability is obtained by adding the terms in the right column.
In passing, note that the probability of obtaining a before is simply
Can you appreciate why obtaining a before is more likely than obtaining a before ?

Example: 8  
Two unit squares are chosen at random from a standard chessboard. What is the probability that the two squares have exactly one corner in common?

Solution: 8  
The total number of ways of selecting two squares from a standard chessboard is simply .
Now, let us find the number of ways of selecting a pair with exactly one corner in common. For that, consider the following figure.
It is immediately evident that the number of favorable ways is .
The required probability is
