Sample Spaces and Events

Roll two dice. What is $P(\text{sum} = 7)$? Before answering, define the machinery. The sample space $\Omega$ is the set of all possible outcomes, here 36 equally likely ordered pairs. An event is a subset of $\Omega$: "sum = 7" is $\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$, six outcomes, so $P = 6/36 = 1/6$.

Getting the sample space right is half of every probability puzzle. Most "paradoxes" come from silently using the wrong $\Omega$: unordered instead of ordered pairs, or outcomes that aren't equally likely.