Probability is favorable outcomes over total outcomes
Every probability sits between 0 (impossible) and 1 (certain), and you can read it as a percentage by multiplying by 100. A fair die has 6 equally likely faces, so the chance of any one number is 1 ÷ 6 = 0.167, or 16.7%.
Combining two events: and, or, not
Most real questions chain events together. Which rule you use depends on whether you mean both, either, or the opposite.
| You want | Rule | Formula |
|---|---|---|
| Not A (complement) | Subtract from 1 | 1 − P(A) |
| A and B, independent | Multiply | P(A) × P(B) |
| A and B, dependent | Multiply with condition | P(A) × P(B | A) |
| A or B | Add, remove overlap | P(A) + P(B) − P(A and B) |
| A or B, mutually exclusive | Add | P(A) + P(B) |
Mutually exclusive means the two events cannot happen at once (an odd roll or an even roll), so their overlap is zero.
Independent vs dependent is the part people get wrong
- Independent. The first outcome does not change the second. Two dice, coin flips, or drawing a card and replacing it. Just multiply the raw probabilities.
- Dependent. The first outcome shifts the odds of the second, so you multiply by the conditional probability. Drawing two hearts from a deck without replacing the first: (13/52) × (12/51) ≈ 0.0588, not (13/52)².
- The "at least one" shortcut. It is usually easier to find the chance of none and subtract. The chance of at least one six in three rolls is 1 − (5/6)³ ≈ 0.421.
Common questions
How do I find the probability of a single event?
Divide the number of favorable outcomes by the total number of equally likely outcomes. Rolling a six on a die is 1 out of 6, which is 0.167 or 16.7%.
What is the probability of A and B both happening?
For independent events, multiply: P(A) times P(B). Two coin flips both landing heads is 0.5 times 0.5 = 0.25. If the events are dependent, use the probability of B given that A already happened instead of P(B).
What is the probability of A or B?
Add the two probabilities and subtract the overlap: P(A) plus P(B) minus P(A and B). You subtract the overlap so the case where both happen is not counted twice. If the events cannot both happen, the overlap is zero and you just add.
What does it mean when a probability is 0 or 1?
Zero means the event is impossible and 1 means it is certain. Everything possible but uncertain falls between the two. Rolling a 7 on a normal die is 0; rolling something from 1 to 6 is 1.
