Basic probability is a single division, but combining two events correctly — the part most people get wrong — depends entirely on which of two different rules applies. Here's the base calculation, how probability differs from odds, and the AND/OR rules with worked examples.
The basic formula
Worked example: the probability of rolling a 6 on a standard six-sided die:
| Calculation | Result |
|---|---|
| 1 ÷ 6 | ≈16.67% |
Another example: a raffle with 1 winning ticket out of 35 total tickets sold:
| Calculation | Result |
|---|---|
| 1 ÷ 35 | ≈2.86% |
Probability vs. odds: not the same number
These get used interchangeably in casual conversation, but they're calculated differently:
| Measure | Formula | 1-in-35 example |
|---|---|---|
| Probability | Favorable ÷ Total | 1 ÷ 35 ≈ 2.86% |
| Odds | Favorable ÷ Unfavorable | 1 to 34 |
Probability compares the favorable outcome to everything (35 total). Odds compare the favorable outcome only to the unfavorable outcomes (34 of them). "1 in 35" and "odds of 1 to 34" describe the exact same underlying situation, just expressed with two different denominators — mixing them up in a sentence changes what's actually being communicated.
Combining two events: the AND rule
When you need the probability of two independent events both happening, multiply their individual probabilities:
Worked example: the probability of flipping heads on a coin and rolling a 6 on a die:
| Calculation | Result |
|---|---|
| 0.5 × (1÷6) | ≈8.33% |
This only works for independent events — ones where the outcome of one doesn't affect the other. The coin flip and the die roll don't influence each other, so multiplying is valid. Dependent events (like drawing two cards from a deck without replacement) need a different, conditional calculation.
Combining two events: the OR rule
For the probability of either of two events happening, the formula depends on whether the events can overlap:
Mutually exclusive example: rolling a 1 or a 6 on a single die (can't get both on one roll):
| Calculation | Result |
|---|---|
| (1÷6) + (1÷6) | ≈33.33% |
Overlapping example: drawing a heart or a king from a standard 52-card deck (the king of hearts is both):
| Calculation | Result |
|---|---|
| (13÷52) + (4÷52) − (1÷52) | ≈30.77% |
Common mistake: simply adding the two probabilities without subtracting the overlap. Without subtracting the 1÷52 for the king of hearts, the king would effectively get counted twice — once as a heart, once as a king — inflating the result to 32.7% instead of the correct 30.77%.
Common mistakes, summarized
- Treating probability and odds as the same number. Probability divides by the total; odds divide by the unfavorable outcomes only.
- Using the AND rule (multiply) on dependent events without adjusting for how the first outcome changes the second event's probability.
- Adding probabilities for overlapping events without subtracting the overlap, which double-counts outcomes that satisfy both conditions.
- Forgetting probability is bounded between 0% and 100%. If a calculation produces a number outside that range, an input or rule was applied incorrectly.
Try the calculator
The probability calculator converts favorable and total outcomes into a percentage, simplified fraction, and odds automatically.
Frequently asked questions
How do you calculate basic probability?
Probability = Favorable outcomes ÷ Total possible outcomes. For example, the probability of rolling a 6 on a standard die is 1 ÷ 6, or about 16.67%.
What is the difference between probability and odds?
Probability compares favorable outcomes to all possible outcomes (e.g., 1 in 6). Odds compare favorable outcomes to unfavorable outcomes (e.g., 1 to 5). They describe the same event but are calculated differently and are not interchangeable numbers.
How do you calculate the probability of two events both happening?
For independent events, multiply their individual probabilities together (the AND rule). For example, the probability of flipping heads and rolling a 6 is 0.5 × (1/6), or about 8.33%.
How do you calculate the probability of either of two events happening?
For mutually exclusive events, add their probabilities together (the OR rule). For events that can overlap, add the two probabilities and then subtract the probability of both happening together, to avoid double-counting the overlap.