The starting point of every probability problem
Before you can say how likely something is, you need to know what could happen. The set of all possible outcomes of a random process is called the sample space. A single die has the sample space {1, 2, 3, 4, 5, 6}. A coin has {heads, tails}. A card drawn from a standard deck has 52 outcomes.
An event is any collection of outcomes you care about, such as “the die shows an even number”, which is {2, 4, 6}. If every outcome in the sample space is equally likely, the probability of an event is the number of outcomes in the event divided by the total number of outcomes. Here that is 3/6 = 1/2.
Everything depends on the list being complete and on the outcomes being genuinely equally likely. Most probability errors come from mistakes at this first stage, long before any arithmetic.
Rules for a good sample space
A usable outcome list has three properties:
- Complete. Every possible result appears somewhere.
- Non-overlapping. Each result matches exactly one entry, so nothing is counted twice.
- Equally likely, if you plan to count. If entries have different chances, you can still use them, but then you must weight them rather than simply count.
The best way to satisfy the third property is to make the outcomes as fine-grained as you need, down to individual dice, individual cards or individual positions.
Two dice: 36, not 11
Roll two dice and ask about the total. The totals run from 2 to 12, which is 11 possible values. It is tempting to say each has a 1 in 11 chance. That is wrong, because the totals are not equally likely.
The way out is to list the underlying outcomes. Treat the dice as distinguishable (say, one red, one blue), so each outcome is an ordered pair. There are 6 × 6 = 36 of them, and each is equally likely. Now count how many pairs give 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 counts add to 36, which is a useful check. A total of 7 can occur in 6 ways, so its probability is 1/6, about 16.7 percent, while 2 or 12 can occur in only one way each, about 2.8 percent. Casino dice games rely on exactly this uneven distribution of totals, and the craps odds are built on the same table.
The common reason for confusion is that two physically identical dice still produce separate outcomes. A roll of a 2 and a 5 can happen two ways (red 2 with blue 5, or red 5 with blue 2), whereas a roll of double 3 can happen only one way. Pretending the dice are labelled, even if they are not, gives the correct counts.
Coins: heads count versus sequences
Flip two coins. If you list outcomes as “0 heads, 1 head, 2 heads”, you have three entries, but they are not equally likely. List the sequences instead: HH, HT, TH, TT. These four are equally likely, so:
- 0 heads: 1 of 4 = 25 percent
- 1 head: 2 of 4 = 50 percent
- 2 heads: 1 of 4 = 25 percent
The principle is the same as with dice. Refine the sample space until the outcomes are symmetrical, then group them into the events you want.
Cards: choose your level of detail
A deck gives you a choice of granularity. If you only care whether a card is an ace, you could use the sample space {ace, not ace}, but its two outcomes have probabilities 4/52 and 48/52, so they are not equally likely and you cannot just count them. Using all 52 cards as the sample space makes every outcome equally likely, and the event “ace” contains 4 of them, giving 4/52 = 1/13.
For hands, the sample space is the set of all unordered hands, which has 2,598,960 members for five cards. Each is equally likely, so the probability of a type of hand is a count divided by that figure. Working out these counts relies on knowing whether order matters, which the combinatorics behind the numbers addresses.
Unequal outcomes are fine if you weight them
Not every game has equally likely outcomes. A loaded die, a weighted spinner, or a loot table with a 1 percent rare item has outcomes with different probabilities. The sample space still exists, but you can no longer count outcomes directly. Instead, you add up the probabilities of the outcomes in an event. For a loot table with listed chances, that is simply what you do, and the listed percentages should add up to 100 percent, which is a quick sanity check. Questions about how rates are published are part of understanding drop rates.
Another option is to split unequal outcomes into equal pieces. A 1 percent item is the same as 1 slot out of 100 equally likely slots, which lets you count again.
Roulette as a simple example
A European-style wheel with 37 pockets (numbers 0 to 36) has a sample space of 37 equally likely outcomes, assuming a fair wheel. A bet on red covers 18 pockets, so its probability is 18/37, about 48.6 percent. A wheel with an extra double-zero pocket has 38 outcomes and the same bet has probability 18/38, about 47.4 percent. The only thing that changed was the sample space. This is the root of the margins behind roulette probabilities.
A checklist before you calculate
- Write down what one trial is.
- List outcomes at the finest level where they look symmetrical.
- Count the total and check it against a product rule: 6 × 6 for two dice, 52 for a card.
- Define your event as a subset of the list and count it.
- If outcomes are not equally likely, switch from counting to adding their probabilities.
- Compare the result with intuition. If a sum of all your probabilities is not 1, something is missing or doubled.
Once the sample space is right, the rest of probability is usually careful counting.