What a probability is
A probability is a number that describes how likely something is to happen. It sits on a scale from 0 to 1. A probability of 0 means the event cannot happen, and a probability of 1 means it is certain. Everything interesting falls in between.
For games, the most useful starting point is this idea: when every outcome is equally likely, the probability of an event is the number of outcomes you are interested in, divided by the total number of possible outcomes.
Take a fair coin. There are two outcomes, heads and tails. One of them is heads, so the probability of heads is 1 out of 2, which is 1/2.
The word “fair” matters. The calculation only works if each outcome really is equally likely. A bent coin or a loaded die would need a different approach, because the outcomes would not share the chance equally.
Three ways to write the same number
The same probability can be written as a fraction, a decimal or a percentage. They are just different costumes.
- Fraction: 1/2
- Decimal: 0.5
- Percentage: 50%
To go from a fraction to a decimal, divide the top by the bottom. To go from a decimal to a percentage, multiply by 100. So 1/4 is 0.25, which is 25%.
Rounding is normal. One in six is 0.1666… as a decimal, which is about 16.7% when rounded to one decimal place. When you see a rounded figure, remember that it is an approximation of the exact fraction.
Counting outcomes with a die
A standard six-sided die has six faces numbered 1 to 6. If it is fair, each face has the same chance on a roll.
- Rolling a 4: 1 outcome out of 6, so 1/6, about 16.7%.
- Rolling an even number: the faces 2, 4 and 6 count, so 3 out of 6, which is 1/2 or 50%.
- Rolling a number greater than 4: the faces 5 and 6, so 2 out of 6, which simplifies to 1/3, about 33.3%.
- Rolling a 7: no face shows 7, so the probability is 0.
Notice that the probabilities of all six single faces add up to 6 × 1/6 = 1. Something always happens, so across a complete list of outcomes the probabilities add to exactly 1.
Counting outcomes with two dice
With two dice, you need to count pairs. Each die has 6 faces, so there are 6 × 6 = 36 equally likely pairs. Only one pair, 1 and 1, makes a total of 2. Six different pairs make a total of 7: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. So the chance of a total of 7 is 6 out of 36, which is 1/6.
This is the reason that some totals turn up more often than others, a topic covered in more detail under two-dice sums. The key lesson for beginners is to count the pairs rather than the totals. There are eleven possible totals from 2 to 12, but they are not equally likely, so you cannot say that each has a 1 in 11 chance.
Counting outcomes with a deck of cards
A standard deck has 52 cards: four suits of 13 cards each. Some quick examples, assuming a well-shuffled deck and a single card drawn:
- A heart: 13 out of 52, which is 1/4 or 25%.
- An ace: 4 out of 52, which simplifies to 1/13, about 7.7%.
- The ace of spades: 1 out of 52, about 1.9%.
- A face card (jack, queen or king): 12 out of 52, which simplifies to 3/13, about 23.1%.
Cards add something that dice do not. After you draw one, the deck changes. The probabilities for the second card depend on what has already come out, an idea that matters enormously in card games.
Probability versus odds
People often use “odds” and “probability” as if they meant the same thing, but they are different ways of expressing chance. Probability compares the outcomes you want with all outcomes. Odds compare the outcomes you want with the outcomes you do not want.
Rolling a 4 on a die has a probability of 1/6. The odds in favour are 1 to 5, because there is one way to succeed and five ways to fail. Converting between the two is a short piece of arithmetic, and it has its own explanation under odds conversion. Shops and betting sites also print their numbers in several styles, which is the subject of odds formats.
Why a probability does not predict the next result
A probability describes a long pattern, not a single outcome. If you flip a fair coin, the chance of heads is 1/2 each time, but that does not mean the next two flips will split evenly. You could see three heads in a row and the coin would still be fair.
Two beginner mistakes follow from this:
- Thinking that results are “due” after a streak. A fair coin has no memory, so a run of tails does not raise the chance of heads on the next flip.
- Thinking that a small number of results must look like the probabilities. Short sequences are often lumpy.
Over many repetitions, the share of heads usually settles closer to one half. That tendency is what the law of large numbers describes, and it says something about proportions over long runs, not about the next flip.
What probability can and cannot do
Probability is a way of describing uncertainty clearly. It lets you compare chances, notice when a claim sounds too good to be true, and understand why a game is built the way it is.
It does not remove the uncertainty. If a game has a house edge, the average result over many plays favours the operator and costs the player money, and no arrangement of bets changes that. Knowing the numbers helps you understand the risk, but it is not a method for winning, and results in any single session can be much better or worse than the average. Gambling can be harmful, and anyone who feels it is becoming a problem should contact a local support service.
A small toolkit to take away
If you remember five things from this guide, make them these:
- A probability is a number between 0 and 1, often written as a fraction, decimal or percentage.
- For equally likely outcomes, divide the favourable outcomes by all outcomes.
- The probabilities of a complete set of outcomes add up to 1.
- Odds and probability describe the same chance in different ways.
- Past results do not change the chance of a fair event happening next.
With those in hand, you can read most introductory probability questions and test them with a few lines of arithmetic. The standard deck is a good place to keep practising, since it has so many easy-to-count events inside it.