What Is a Sample Space?

The grid, and what each cell is worth

The point of a sample space is not neatness. It is that once every outcome is listed and the outcomes are equally likely, a probability becomes a matter of counting cells and dividing by 36.

Students who answer 1/11 for a total of 7 are not making an arithmetic mistake. They are treating a list of answers as if it were a list of outcomes, and the two are only the same thing when every answer arises in exactly one way. Rolling two dice, that is true only for 2 and for 12.

Three questions, one grid, no recounting

Once the grid is drawn, further questions cost nothing. Each is a different shape shaded on the same 36 cells.

This is why it is worth drawing the grid even when the first question looks easy enough to do in your head. Papers rarely ask only one thing about a pair of dice.

• P(total is 7): the cells 1+6, 2+5, 3+4, 4+3, 5+2, 6+1 — a diagonal of 6 cells, giving 6/36 = 1/6.
• P(at least one 5): a complete row and a complete column, sharing the cell (5, 5) — 6 + 6 − 1 = 11 cells, giving 11/36.
• P(the two numbers differ by 1): the pairs (1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5) — two short diagonals of 5, giving 10/36 = 5/18.

When a list is not a sample space

Two conditions have to hold. Every possible outcome must be there, and no outcome may appear twice. A third condition — that the outcomes are equally likely — is not part of the definition but is required before you can find a probability by counting.

The classic failure is dropping the order. With two identical dice it feels natural to write (2, 5) once, which gives 21 outcomes instead of 36. But the dice do land in a particular way whether or not you can tell them apart, and (2, 5) genuinely happens twice as often as (5, 5). The same mistake with two coins produces three outcomes — two heads, two tails, one of each — and the false conclusion that one of each has probability 1/3 rather than 2/4.