Sample Spaces

Before you can find a single probability, you have to answer one question first: what could possibly happen? Roll a die and you know it in your bones — a 1, 2, 3, 4, 5 or 6. That complete list of everything that could come out is called the sample space.

It sounds almost too simple to bother naming. But the moment two things happen at once — two dice, or a coin and a spinner — the possibilities multiply and it becomes horribly easy to miss one, or to count the same thing twice. The whole secret to getting probabilities right is to organise the sample space so you can count it without slipping up. A neat list, a grid, or a table turns a confusing muddle into a simple tally.

P(\text{event}) = \frac{\text{favourable outcomes}}{\text{total outcomes in the sample space}}

Every probability is really just two counts — and both of them are read straight off the sample space. Get the sample space right and the probability falls out on its own.

Listing a small sample space: two coins

Toss a 10p and a 20p at the same time. Each coin lands Heads (H) or Tails (T), so together there are four possibilities. Because the coins are different, we track them in order — first coin, then second:

\{\, HH,\ HT,\ TH,\ TT \,\}

Four outcomes, each equally likely. Now any question is just counting. What is the chance of getting exactly one head? Two of the four outcomes fit — HT and TH — so

P(\text{exactly one head}) = \frac{2}{4} = \frac{1}{2}.

Notice that HT and TH are different outcomes, even though both are "one head and one tail". Keeping them separate is exactly what makes the count come out right.

To work out a probability when two events happen together:

The grid for two dice

With two dice the list gets long — 36 outcomes — so a plain list is clumsy. A two-way sample space diagram is far better: die A runs down the rows, die B across the columns, and every cell is one combined outcome. The grid is 6 \times 6, which is

6 \times 6 = 36 \text{ outcomes.}

Step through to fill in the sums, then to spot every way of making a total of 7. Count the highlighted cells and the probability is immediate.

Six cells light up, so P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6} \approx 0.17. No cleverness, no formula to memorise — just count the favourable cells and divide by 36.

Combining different things: a spinner and a coin

The two events don't have to match. Spin a spinner with three colours — Red, Green, Blue — and toss a coin (H or T). A little table lays out the whole sample space: three colours across, two coin faces down.

\begin{array}{c|ccc} & R & G & B \\ \hline H & RH & GH & BH \\ T & RT & GT & BT \end{array}

That is 3 \times 2 = 6 outcomes. Want the chance of Blue and Heads? Exactly one cell fits, so

P(\text{Blue and Heads}) = \frac{1}{6}.

And the chance of Heads with any colour? The whole top row — 3 cells — so \tfrac{3}{6} = \tfrac{1}{2}, just as you'd hope for a fair coin.

One grid, many questions

The beauty of a sample space diagram is that one picture answers a whole family of questions. Keep the two-dice grid in mind and count the favourable cells for each:

Same 36 outcomes every time; only the count of favourable cells changes. That is the whole method — build the sample space once, then answer anything by counting.

The classic blunder is to list the sample space haphazardly and either miss an outcome or count one twice. The cure is to be systematic — march through it in a fixed order (all the first-die-1 outcomes, then all the first-die-2 outcomes, and so on) so you can see that nothing is skipped.

The sneakiest trap of all: for two distinguishable dice, (2,5) and (5,2) are different outcomes — die A shows 2 and die B shows 5 in one, the other way round in the other. That is why there are 36 outcomes, not 21. If you lazily treat "a 2 and a 5" as a single outcome you'll get 21 and every probability afterwards will be wrong. When you can tell the two dice apart, order matters.

Glance at the two-dice grid and something jumps out: the total 7 runs right down the whole diagonal — (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — six ways. Meanwhile 2 happens only as (1,1) and 12 only as (6,6) — one way each, the rarest of all.

Seeing the whole sample space at once turns a confusing probability into an obvious count. This is no accident of gambling folklore: the game of craps and countless board games are designed around the fact that 7 is the most common roll and 2 and 12 the least. The grid doesn't just answer the question — it shows you why the answer had to be what it is.

See it explained