Outcomes, sample spaces and events
Probability starts with a list of everything that could happen. Each single result is an outcome, and the full list is the sample space. For one six-sided die the sample space is 1, 2, 3, 4, 5 and 6. For a standard 52-card deck, drawing one card has 52 possible outcomes.
An event is any collection of outcomes you care about. “Roll an even number” is the event {2, 4, 6}. “Draw a heart” is a set of 13 cards. Events are the questions you ask; outcomes are the raw material that answers them.
When every outcome is equally likely, the probability of an event is a simple ratio:
probability = (number of outcomes in the event) / (total number of outcomes)
So the chance of an even roll is 3/6 = 1/2, and the chance of drawing a heart is 13/52 = 1/4. The “equally likely” condition matters. A loaded die, a weighted spinner or a loot table with unequal rarities breaks it, and then the ratio no longer applies directly.
The scale from 0 to 1
Probabilities run from 0 (cannot happen) to 1 (certain). You can write them as fractions, decimals or percentages: 1/4, 0.25 and 25% are the same thing. Rolling a 7 on one standard die has probability 0, because 7 is not in the sample space. Rolling a number from 1 to 6 has probability 1.
Two things follow. First, the probabilities of all the outcomes in a sample space add up to 1. Second, the chance an event does not happen is 1 minus the chance it does. That second idea turns out to be so useful that it has its own treatment under the complement rule.
A probability describes what happens over many repetitions or how much confidence a fair model supports. It does not say what the next single trial will do.
Independent events
Two events are independent when knowing that one happened tells you nothing about the other. Successive dice rolls are the standard example: the die has no memory. Believing otherwise, for instance that a number is “due” after a long absence, is the gambler’s fallacy, and it is one of the most common errors in games of chance.
Draws from a deck without putting cards back are different. Once an ace has left the deck, the cards that remain are a different mix, so later draws depend on earlier ones. That situation is covered in more depth under dependent events.
The multiplication rule
For independent events, the chance that both happen is the product of their chances.
Illustration: roll two dice. The chance both show a six is 1/6 × 1/6 = 1/36. The chance the first die is even and the second is a 5 or 6 is 1/2 × 2/6 = 2/12 = 1/6.
For dependent events, you multiply the chance of the first event by the chance of the second given that the first happened. Draw two cards from a standard deck without replacement. The chance both are aces is 4/52 × 3/51 = 12/2652 = 1/221. The second fraction has changed from 4/52 to 3/51 because one ace and one card in total are gone.
Use “and” as the cue. When a question needs one thing and another, you are almost always multiplying.
The addition rule
For events that cannot happen together (mutually exclusive events), the chance that one or the other happens is the sum of their chances.
Illustration: one die roll. The chance of a 1 or a 2 is 1/6 + 1/6 = 2/6 = 1/3. These two results cannot both occur on a single roll, so simple addition works.
When events can overlap, adding double counts the overlap, so you subtract it once:
P(A or B) = P(A) + P(B) − P(A and B)
Illustration: draw one card. The chance it is a heart or a king is 13/52 + 4/52 − 1/52 = 16/52 = 4/13. The king of hearts belongs to both groups, so it is subtracted once. Forgetting this step is a classic case of adding probabilities wrongly. For larger overlaps, the inclusion-exclusion principle generalises the same idea.
Use “or” as the cue: one thing or another usually means adding, with a check for overlap.
A common trap: adding when you should not
It is tempting to say that the chance of at least one six in two dice rolls is 1/6 + 1/6 = 1/3. That is wrong, because the two events can both happen (two sixes), and the overlap has been counted twice. The correct figure is 11/36, about 30.6%. One route to it is 1/6 + 1/6 − 1/36 = 12/36 − 1/36 = 11/36. Another is to find the chance of no sixes, (5/6) × (5/6) = 25/36, and subtract from 1.
If you ever add probabilities and the total for a repeated trial climbs towards or past 100% after enough tries, that is a signal something is off. Six rolls do not make a six certain.
Putting the rules together
Most real questions combine the rules. Consider: what is the chance that a card drawn from a deck is a heart, and a die roll is then a 6? The two actions are independent, so multiply: 1/4 × 1/6 = 1/24.
A sensible habit is to write the question in plain words, mark each “and” and “or”, decide whether the pieces are independent, and only then do any arithmetic. It also helps to sanity check the final number: it should lie between 0 and 1, and it should be smaller than the chance of either piece alone when you used “and”.
From probability to decisions
Probabilities become useful for decisions when paired with payoffs, which leads to expected value. In casino games the payoffs are set so that the long-run average is negative for the player, a feature usually called the house edge. Knowing the maths explains why, but it does not create a way to overcome it. Outcomes in the short run are uncertain and can swing widely, and over the long run the law of large numbers pulls averages toward the theoretical figure, not in the player’s favour.
Gambling can be harmful. Only spend what you can afford to lose, never chase losses, and if gambling is causing you or someone you know difficulty, contact a local support service.
Where to go next
The rules above are the whole foundation. Everything more advanced, such as conditional probability, distributions or simulation, is built from outcomes, events, independence, and the addition and multiplication rules. If you keep the cues “and means multiply, or means add, check for overlap and dependence”, you will handle the large majority of dice and card questions correctly.