Where the numbers come from
Most game mechanics that involve chance boil down to the same method: list the equally likely outcomes, count the ones you care about, and divide. The examples below apply that method to a die pair, a deck and a spinner. All numbers assume fair equipment.
Two dice: a table of totals
Two six-sided dice give 6 × 6 = 36 equally likely pairs. Count how many pairs make each total:
| Total | Pairs | Probability |
|---|---|---|
| 2 | 1 | 1/36 |
| 3 | 2 | 2/36 |
| 4 | 3 | 3/36 |
| 5 | 4 | 4/36 |
| 6 | 5 | 5/36 |
| 7 | 6 | 6/36 |
| 8 | 5 | 5/36 |
| 9 | 4 | 4/36 |
| 10 | 3 | 3/36 |
| 11 | 2 | 2/36 |
| 12 | 1 | 1/36 |
The pairs column adds up to 36, which is a good check. The middle total, 7, has the most pairs, and totals fall away symmetrically on either side. The reasons behind this shape are explored under two-dice sums, and the same principles for tabletop games appear in the dice probability article.
Some questions that follow from the table:
- Total of 8 or more: 5 + 4 + 3 + 2 + 1 = 15 pairs, so 15/36 = 5/12, about 41.7%.
- Total of 6 or less: 1 + 2 + 3 + 4 + 5 = 15 pairs, also 5/12.
- Exactly 7: 6/36 = 1/6, about 16.7%.
These add up sensibly: 15 + 15 + 6 = 36.
At least one six on two dice
A common trap is to say that the chance of at least one six on two dice is 1/6 + 1/6 = 1/3. That counts the double six twice and overstates the chance.
An easier route is to count the opposite. The chance a single die is not a six is 5/6. For two independent dice, the chance neither is a six is 5/6 × 5/6 = 25/36. So the chance of at least one six is 1 − 25/36 = 11/36, about 30.6%. Direct counting agrees: there are 11 pairs containing at least one six (5 where the first die is six and the other is not, 5 where the second is six and the other is not, plus the double six). This approach is the subject of complement counting.
Card draws
A standard deck has 52 cards in four suits of 13. For one draw from a shuffled deck:
- Any heart: 13/52 = 1/4, so 25%.
- Any face card (jack, queen, king): 12/52 = 3/13, about 23.1%.
- A red card that is a face card: 6/52 = 3/26, about 11.5%.
Many card games deal several cards, and then you need to account for the cards that have already left the deck. For two cards drawn in a row, the chance that both are hearts is 13/52 × 12/51 = 156/2652 = 1/17, about 5.9%. The second fraction is smaller because the first heart is no longer available, the central idea in without replacement. How the whole deck’s structure shapes every game is covered in the article on the standard deck.
Spinners
A spinner is simply a wheel divided into sections. If the sections are the same size, the chance of landing on one is the number of those sections divided by the total.
Imagine an eight-section spinner with 3 red, 2 blue, 2 green and 1 yellow section. The probabilities are:
- Red: 3/8 = 37.5%
- Blue: 2/8 = 1/4 = 25%
- Green: 2/8 = 1/4 = 25%
- Yellow: 1/8 = 12.5%
These add to 37.5 + 25 + 25 + 12.5 = 100%, as they should.
The spinner also shows how a designer can set different chances. If a rare event should happen about one time in eight, give it one section out of eight. If the sections are not the same size, you would use the share of the circle each takes up rather than counting them.
Expected results in a simple game
Suppose a made-up board game lets you spin the spinner above and gain points: red gives 1 point, blue gives 2, green gives 3 and yellow gives 8. The average points per spin are:
(3/8 × 1) + (2/8 × 2) + (2/8 × 3) + (1/8 × 8) = 3/8 + 4/8 + 6/8 + 8/8 = 21/8 = 2.625 points.
This weighted average is the idea behind expected value. It tells you what happens on average across a very large number of spins, not what the next spin will give. Over just a few turns, a player can score well above or below 2.625 per spin.
What the numbers do and do not tell you
The mathematics is reliable for describing fair devices. It is not a promise about any particular session. Knowing that a total of 7 is the single most common roll does not mean it will come up on a schedule, and past rolls do not influence the next ones.
In games where money is wagered, rules are generally set so the operator keeps an average advantage, and that means players lose money in the long run. The arithmetic here is for understanding how games work and for designing fair play, not for finding a way to win. If gambling is causing you stress or financial trouble, a local support service can help.