How to Figure Out Probability: A Step-by-Step Method With Dice, Roulette and Mines Examples

To figure out probability, count the outcomes you want, count all the possible outcomes, and divide the first by the second. That works whenever each outcome is equally likely. Red on a European roulette wheel is 18 red pockets out of 37, so its probability is 18 ÷ 37 = 48.65%. Harder questions, like "at least one" or "two in a row", use a few extra rules that build on the same idea.

Below is a six-step method that covers almost every probability question you'll meet at a casino table or in a game, with worked examples from dice, roulette and Mines. Each step comes with a formula you can reuse. If you'd rather skip the arithmetic, our chance calculator does the "at least once" and losing-streak math for you.

The six steps at a glance

1. Define the experiment and list the outcomes.
2. Check the outcomes are equally likely.
3. Divide favourable outcomes by total outcomes.
4. Use the complement for "at least one".
5. Multiply for "and".
6. Count with combinations, then sanity-check.

Step 1: Define the experiment and list every outcome

Be precise about what you're doing once. "Spin a European roulette wheel" has 37 outcomes: the numbers 0 to 36. "Roll one six-sided die" has 6. "Roll two dice" has 36 if you treat the dice as different (first die, second die), because 6 × 6 = 36. The full list is called the sample space.

Getting this step wrong is the most common source of errors. "Roll two dice and look at the total" has 11 possible totals (2 to 12), but those 11 totals are not the outcomes you should count, as step 2 explains.

Step 2: Check the outcomes are equally likely

The divide-and-count method only works if every outcome in your list has the same chance. Each pocket on a fair wheel and each face of a fair die qualifies. Dice totals don't: a total of 7 can be made six ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), while 12 can only be made one way (6+6).

The fix is to list outcomes at a finer level, the 36 ordered pairs, where every outcome is equally likely, and then group them. Our craps odds guide shows the full two-dice table built this way.

Step 3: Divide favourable outcomes by total outcomes

This is the core formula. Britannica defines probability in gambling as the number of favourable outcomes divided by the total number of possible outcomes:

More examples:

When favourable outcomes can't happen together, you can also add their probabilities: red or zero on a European wheel is 18/37 + 1/37 = 19/37 = 51.35%. That's the same as counting 19 favourable pockets.

Step 4: Use the complement for "at least one"

Questions with "at least one" have many ways to succeed (once, twice, three times…). It's far easier to work out the one way to fail, which is "never", and subtract:

The complement also explains why "1 in N" doesn't mean "within N tries". One roulette number in 37 spins: P(never) = (36/37)^37 = 36.3%, so the number appears at least once only 63.7% of the time.

Step 5: Multiply for "and"

When several things must all happen, multiply. If the events are independent (one result doesn't affect the next), OpenStax gives the multiplication rule:

When events are not independent, multiply the chance of each step given the steps before it. Mines is the classic case: each safe tile you uncover is gone, so the next pick has fewer tiles left.

Spins and dice rolls are independent; draws from a fixed set without putting anything back (cards, Mines tiles, keno balls) are not. Treating independent events as if they were linked, so that a loss makes a win "due", is the gambler's fallacy.

Step 6: Count with combinations, then sanity-check

When outcomes are too many to list, count them. The OpenStax counting principles give three tools:

The Mines example from step 5 again, this time by counting:

Then sanity-check. Every probability is between 0 and 1. The probabilities of all outcomes add up to 1. And the answer should match what happens over many trials, because probability is the long-run relative frequency of an outcome, as OpenStax puts it.

Checking the method against our simulations

The best test of a probability calculation is a long run of random trials. Our data includes both exact figures and simulations for dice, roulette and Mines:

The simulated results land within a few hundredths of a percentage point of the counted ones. That's the law of large numbers at work: over a million trials, observed frequencies settle very close to the true probability.

From probability to the price of a bet

Once you can figure out probability, you can price any bet. Multiply the chance of winning by the total return and compare with 1:

Our odds vs probability guide explains how payout odds and true odds relate, the roulette house edge guide applies the formula to every wheel, and which casino games have the best odds compares edges across games. For crypto dice, the dice odds calculator turns any win chance into a multiplier.

Common mistakes

• Counting outcomes that aren't equally likely. Two-dice totals 2 to 12 are 11 outcomes, but 7 is six times as likely as 2.
• Adding when you should multiply. Two 6s in a row is 1/6 × 1/6, not 1/6 + 1/6.
• Forgetting removed items. In Mines, cards and keno, each draw changes what's left.
• Expecting "1 in N" within N tries. One roulette number in 37 spins shows up only about 64% of the time.
• Thinking past results change the next one. Independent spins and rolls have no memory.

More guides on odds, house edge and betting systems are on our odds and strategy math hub.