Counting the ways
Two dice, one question: probability = ways it can happen ÷ ways anything can happen.
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Roll a pair of dice — one roll, two faces. What's the chance they add up to seven? The gut says no idea: dice are random. But random is not the same as uncountable. Every roll comes from a fixed menu of outcomes, and a menu can be counted.
So write the menu down. The first die can land six ways — one row for each face. The second die can land six ways too — one column for each. Six rows by six columns: thirty-six squares, one for every pair of faces. Fair dice play no favorites, so all thirty-six squares are equally likely.
Now label every square with its sum, and hunt for sevens. One and six. Two and five. Three and four. Plus their mirror twins: six and one, five and two, four and three. They line up on a diagonal — exactly six squares. Six sevens out of thirty-six equally likely squares, so the chance is six over thirty-six: one in six.
Not every total is so rich. A two needs snake eyes, a one on each die — a single square in the whole grid. A twelve needs double sixes — one square again. A seven owns six. The dice don't favor seven; middle sums simply have more ways to happen, and more ways means more likely.
Probability is the ways your result can happen, divided by the ways anything can happen. A seven claims six of thirty-six equally likely squares: one in six. The gut had no answer, because a gut can't hold thirty-six squares at once. A grid can. Chance isn't a feeling — it's a count: the ways you want, over the ways there are.