This is a reference list. Each idea gets a short definition and a small example, so you can recognise the term when you meet it elsewhere. The examples use simple, made-up numbers, and nothing here is a way to win money.
1. Sample space
The sample space is the complete list of things that could happen. For one roll of a die it is the six faces. For one coin flip it is heads and tails. Before you can work out any chance, you need to know what is on this list and whether its items are equally likely.
2. Probability
Probability is a number from 0 to 1 that says how likely an event is. When outcomes are equally likely, it is the number of favourable outcomes divided by the total. Drawing a heart from a full deck is 13 out of 52, which is 1/4.
3. Odds
Odds compare the ways an event can happen with the ways it cannot. A heart has odds of 13 to 39 in favour, which simplifies to 1 to 3. Odds and probability describe the same chance, but the numbers look different, so it helps to know which one a source is using.
4. Independent events
Two events are independent when one does not change the chance of the other. Two flips of a fair coin are independent: the first result tells you nothing about the second. For independent events, the chance of both happening is found by multiplying. Two heads in a row is 1/2 × 1/2 = 1/4.
5. Conditional probability
Conditional probability is the chance of something given that you already know something else. If one heart has been taken from a full deck and not replaced, the chance that the next card is a heart is 12 out of 51, because the deck has changed. Conditional probability is how new information updates a chance, and it has a full explanation under conditional probability.
6. Mutually exclusive events
Events are mutually exclusive when they cannot both happen on the same trial. A single die cannot show both a 2 and a 5. Because they never overlap, you can add their probabilities: 1/6 + 1/6 = 2/6 = 1/3 for rolling a 2 or a 5.
7. Expected value
Expected value is the average result you would get per play if you could repeat the same situation a very large number of times. Take a made-up game where a fair coin pays 3 units on heads and costs 2 units on tails. The expected value is (1/2 × 3) − (1/2 × 2) = 1.5 − 1 = 0.5 units per play. Real games rarely look like this, and a positive or negative average says nothing certain about the next play.
8. Variance
Variance describes how spread out results are around the average. Two games can share the same expected value, but one may pay small amounts often while the other pays large amounts rarely. The second one has higher variance, so its results swing more. A related measure, the standard deviation, puts that spread in the same units as the results themselves.
9. The law of large numbers
The law of large numbers says that, as you repeat a random process more and more, the average of the results tends to move toward the expected value. It is about averages over many trials. It does not say that early imbalances get corrected, and it cannot tell you what will happen on any single trial.
10. The gamblers fallacy
The gamblers fallacy is the belief that a streak makes the opposite result more likely. After five tails in a row, a fair coin is still 50% likely to land heads on the next flip, because the coin has no memory. The related belief that streaks will continue is discussed under hot hand belief. In both cases, the mistake is treating independent events as if they were linked.
A bonus idea: house edge
Many casino-style games are built so that the operator keeps a small average share of the money wagered, known as the house edge. Because that edge works in the operator’s favour every round, the average outcome for players over many plays is a loss, even though individual sessions can end in a profit. Understanding this is part of understanding the risk, not a route to beating it.
How the ideas fit together
The ideas build on each other. The sample space gives you the outcomes. Probability and odds put numbers on them. Independence and conditional probability tell you how events affect one another. Expected value and variance summarise the average and the spread, while the law of large numbers and the gamblers fallacy explain what long runs do and do not tell you.
If you only have time to learn two things, learn that past results do not change the odds of a fair, independent event, and that an average result over many plays says little about a short session. Gambling can lead to real harm, and anyone who finds it hard to stop should reach out to a local support service.