That list has a name — the sample space — and once you have it, probability stops being magic and turns into plain counting.
Start hereA sample space is the complete list of every result that could possibly happen. Write that list down and the scary question "what are the chances?" becomes a tame one: how many of these results am I hoping for, out of how many there are in total?
That's the entire trick of this page. Make the full list, count the ones you want, and the probability falls right out as a fraction. No spells, no luck — just careful counting on a tidy list.
Three little words do all the heavy lifting in probability, so let's pin them down before anything moves. An outcome is one single thing that can happen — one roll landing on a 4, one coin showing heads. An event is something you care about, which might cover several outcomes at once — like "rolling an even number," which is really the outcomes 2, 4, or 6 bundled together. And the sample space is the grand list of every outcome that's even possible.
Start with the simplest gadget in the world: one ordinary six-sided die. Roll it, and the result has to be one of six faces — nothing else is possible. So its sample space is just this short list:
Now picture the event "rolling an even number." Which outcomes count? The 2, the 4, and the 6 — that's 3 of the 6. So the probability is 3/6, which tidies up to 1/2. Notice what you actually did: you didn't guess and you didn't feel it out. You listed all six outcomes, circled the ones that fit, and made a fraction. That's the whole method, and it never changes — it just gets a bigger list.
A quiet but crucial assumption hides here: on a fair die, all six faces are equally likely. That's the only reason we're allowed to count outcomes and turn the count straight into a probability. Keep that in your back pocket — it'll matter a lot later.
Before we wheel out the dice, let's flip two coins — a first coin and a second coin — and build their sample space by hand. Each coin can land heads (H) or tails (T), so together they can land in four ways. Here's the rule worth tattooing on your brain: list every outcome, then count the ones you want.
Tap a condition below and the matching outcomes light up, with a live count underneath. Watch what happens to "exactly one head" especially — it's a classic trap.
Exactly one head means HT or TH — two different outcomes.
Here's the trap, spelled out. People often say "two coins can give two heads, two tails, or one of each — three things, so one-of-each must be 1/3." Not quite. HT and TH are different outcomes: heads-then-tails is not the same flip as tails-then-heads. Keeping them separate gives four equally likely outcomes, so "exactly one head" is 2 of 4 = 1/2, not 1/3. The moment your list is honest, the counting is honest too.
Two coins gave 2 × 2 = 4 outcomes. Two dice work exactly the same way, just bigger: the first die has 6 faces and the second die has 6 faces, and every face of the first can pair with every face of the second. So the sample space holds 6 × 6 = 36 outcomes — pairs like (1, 1), (1, 2), all the way to (6, 6).
You could try to scribble all 36 pairs in a long sentence. Please don't — that's exactly where mistakes breed. Rattle them off in a line and you'll skip one or write another twice, and your counting is wrong before you've even started. The fix is to stop writing a list and start drawing a grid.
Picture a 6 × 6 table. The rows are the first die (1 to 6, top to bottom) and the columns are the second die (1 to 6, left to right). Every little square sits at one row and one column, so it stands for exactly one pair — and there are 6 × 6 = 36 squares. Nothing is missing, because every row meets every column. Nothing is doubled, because each pair has its own home. The grid is the whole sample space, organised so neatly that counting becomes almost impossible to get wrong.
To feel the difference, just read the top row aloud: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) — the six rolls where the first die shows a 1. The next row is the same six pairs starting with a 2, and so on down to the row that starts with a 6. Six rows of six. That's the kind of perfect, complete bookkeeping a scribbled sentence almost never manages — and it's why we'll do all our counting on the grid from here on.
The whole sample space is below: 36 squares, one for each pair. Pick a condition and the squares that fit light up — with the exact probability counted live, straight off the grid.
Six squares add up to 7 — they run along a neat diagonal.
Let's count one fully by hand so you trust the machine. Take the event "the total is 4." Hunt the grid for squares where the two dice add to 4 and you'll find exactly three: (1, 3), (2, 2) and (3, 1). Three squares out of thirty-six, so the probability is 3/36 = 1/12. You didn't need a formula — you found the matching outcomes and made a fraction, the same three moves every single time.
Try them all and a pattern jumps out. "Sum = 7," "a double," and "sum > 9" each light up 6 squares — so all three have the same probability, 6/36 = 1/6, even though they look nothing alike. "At least one 6" is the sneaky one: a whole row plus a whole column is 12 squares, but the corner square (6, 6) belongs to both, so you'd be counting it twice. The honest count is 12 − 1 = 11 squares — 11/36. The grid catches that double-count for you; a hurried list never would.
When you add two dice the total can be anything from 2 to 12. But here's the thing players have known for centuries: a 7 turns up far more often than a 2 or a 12. The grid shows you exactly why. There's only one way to make a 2 — both dice showing 1 — but there are six different ways to make a 7. More squares means more chances, and 7 sits on the longest diagonal of all.
Each bar below counts how many of the 36 squares give that total. Pick a sum, or tap a bar, and see its odds.
A 7 can be made six ways — the tallest bar of all.
The shape is a tidy pyramid: rare totals at the edges (2 and 12, one way each), the common ones piled up in the middle, peaking at 7. That's why board games make 7 special, and why "snake eyes" (a 2) feels lucky when it lands — it's genuinely the rarest roll, sharing last place with boxcars, the double 6.
Almost every mistake in this topic comes from building the list wrong. Here are the two big ones — and why the grid quietly saves you from both.
"The total can be 2, 3, 4, … up to 12 — that's 11 possible sums, so each one has a probability of 1/11."
The 11 sums are not equally likely, so you can't just count them. The equally likely things are the 36 outcomes on the grid — not the totals. A 7 happens 6 ways (6/36) while a 2 happens only 1 way (1/36). Always count from the real sample space, never from a list of answers.
"(1, 2) and (2, 1) are really the same roll, so I'll count them once. There are only 21 outcomes, not 36."
Treat the two dice as a first die and a second die — then (1, 2) and (2, 1) are genuinely different outcomes, and there are 36. This isn't fussiness: only by keeping them separate do all the outcomes stay equally likely, which is the one thing that lets you count at all. Merge them and your fractions go wrong.
Quick gut-check for both traps: ask "are the things I'm counting equally likely?" If yes, count away. If no (like the 11 totals), drop down to the real sample space — the one where every outcome has the same fair shot.
Four quick rounds. For each one, picture the sample space — the four coin outcomes or the 36-square grid — count the matches, and pick the fraction. Tap an answer to lock it in and see the working.
Write the sample space — every outcome that could happen, each one equally likely.
Count the outcomes that match the event you care about. A grid beats a sentence every time.
Make the fraction: matches ÷ total. That fraction is the probability. Done.