The rules in one place
Almost every introductory probability question uses a short list of rules. Learn these four and you can solve a large share of everyday game questions. Each is shown with a small worked example, and the numbers are all illustrative.
Rule 1: probabilities stay between 0 and 1, and the whole adds to 1
A probability of 0 means impossible, and 1 means certain. If you list every possible outcome, their probabilities add up to exactly 1. This gives a free check: if your answers do not total 1, something is off.
For a fair die, each face is 1/6, and six faces add to 1.
Rule 2: the complement rule
The chance that an event does not happen is 1 minus the chance that it does:
P(not A) = 1 − P(A)
If the chance of drawing a heart is 1/4, the chance of not drawing a heart is 1 − 1/4 = 3/4.
This rule is especially useful for “at least one” questions. What is the chance of at least one head in three coin flips? Rather than listing every combination with one, two or three heads, work out the single case you do not want: no heads at all, meaning three tails in a row. That is 1/2 × 1/2 × 1/2 = 1/8. So the chance of at least one head is 1 − 1/8 = 7/8, which is 87.5%. The method is explored further under complement counting.
Rule 3: the addition rule for “or”
If two events cannot happen together (they are mutually exclusive), add their probabilities:
P(A or B) = P(A) + P(B)
If they can overlap, subtract the overlap once so it is not counted twice:
P(A or B) = P(A) + P(B) − P(A and B)
Worked example with a die: what is the chance of rolling an even number or a number above 4? Even: 3/6. Above 4: 2/6. The face 6 is in both, 1/6. So 3/6 + 2/6 − 1/6 = 4/6 = 2/3. Counting directly, the faces 2, 4, 5 and 6 qualify, which is four faces out of six, and it matches.
Handling more than two overlapping events is a larger version of the same idea, covered under inclusion-exclusion principle.
Rule 4: the multiplication rule for “and”
To find the chance that two events both happen, multiply. How you multiply depends on whether the events affect each other.
Independent events. When the first result does not change the second, multiply the separate probabilities:
P(A and B) = P(A) × P(B)
Two dice, both showing a six: 1/6 × 1/6 = 1/36, about 2.8%.
Dependent events. When the first result changes the second, use the chance of the second given the first:
P(A and B) = P(A) × P(B given A)
Cards give the clearest case. Draw two cards from a shuffled deck without replacing the first. What is the chance of a red card followed by a black card?
- First card red: 26/52 = 1/2.
- Second card black, given the first was red: 26 black cards remain among 51, so 26/51.
- Both: 26/52 × 26/51 = 676/2652 = 13/51, about 25.5%.
If you put the first card back and reshuffle, the second draw would be 26/52 again, giving 1/2 × 1/2 = 1/4 = 25%. The small difference between 25% and about 25.5% comes from the missing card. The deeper explanation of how conditions change a chance is under conditional probability, and a tidy way to track multi-step cases is the probability tree.
Putting the rules together
Real questions often use several rules at once. Suppose you roll two dice, and you want the chance that the first die is a six or the second die is a six, which is the same as at least one six.
- Method A (complement): the chance neither die is a six is 5/6 × 5/6 = 25/36. So the chance of at least one six is 1 − 25/36 = 11/36.
- Method B (addition with overlap): 1/6 + 1/6 − 1/36 = 6/36 + 6/36 − 1/36 = 11/36.
Both methods give 11/36, about 30.6%. Having two routes to the same answer is a useful way to check your work.
Counting the possible ordered outcomes in a multi-stage setup is the foundation of the multiplication principle, which gives a different view of why the multiply rule works.
Choosing the rule
| Question wording | Usual rule |
|---|---|
| “not”, “none”, “at least one” | Complement |
| “A or B” | Add, subtract any overlap |
| “A and B”, unrelated events | Multiply |
| “A and B”, one changes the other | Multiply, adjusting the second chance |
A note on using these rules for games of chance
These rules explain the chances in a game. They do not offer a way to beat a game with a built-in disadvantage for the player, and they do not predict the next result. Fair devices have no memory, so a past streak does not alter the next roll or draw. If a game ever feels stressful rather than fun, taking a break and speaking to a local support service is a sensible step.