What this calculator does
Combining probabilities correctly depends on whether events are independent and whether they can overlap. The rule for "and" multiplies; the rule for "or" adds and then subtracts the overlap, because otherwise the cases where both occur get counted twice.
The most practically useful line is the last: the chance of at least one occurrence across repeated trials. It is computed as one minus the chance of never occurring, and it explains why unlikely events keep happening: enough attempts make almost anything probable.
The formula
For independent events, the probability that both occur is the product of the individual probabilities. The probability that either occurs adds them and subtracts the overlap. The probability of neither multiplies the two complements, and one minus that gives at least one occurrence across repeated trials.
| Term | Meaning |
|---|---|
| P(A) | The probability of event A. |
| Independent | One event does not affect the probability of the other. |
| Complement | The probability an event does not occur: 1 − P. |
| Odds | The ratio of favourable to unfavourable outcomes. |
The inputs explained
| Field | What to enter |
|---|---|
| P(A) (%) | The probability of event A, as a percentage. |
| P(B) (%) | The probability of event B, as a percentage. |
| Repeat A this many times | How many times event A is repeated, for the at-least-once calculation. |
When to use it
Assessing compound risk
Two independent things must both go right for a plan to work. Multiplying their probabilities usually gives a lower number than intuition expects, especially as the number of dependencies grows.
Understanding repeated exposure
A 1 per cent risk per attempt becomes a 63 per cent chance of occurring at least once across a hundred attempts. This is why small per-event risks matter at scale.
Converting to odds
The odds line translates a percentage into the form used in betting and clinical reporting: a 25 per cent probability is odds of 1 in 4.
Checking whether events are independent
The calculations here assume independence. If one event influences the other, the "and" result will be wrong, and conditional probability is required instead.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Combining two probabilities
Event B held at 40 per cent while event A varies.
| P(A) | P(A and B) | P(A or B) | P(neither) | P(exactly one) |
|---|---|---|---|---|
| 5% | 2.00% | 43.0% | 57.0% | 41.0% |
| 10% | 4.00% | 46.0% | 54.0% | 42.0% |
| 25% | 10.0% | 55.0% | 45.0% | 45.0% |
| 50% | 20.0% | 70.0% | 30.0% | 50.0% |
| 75% | 30.0% | 85.0% | 15.0% | 55.0% |
| 90% | 36.0% | 94.0% | 6.00% | 58.0% |
How repetition builds up risk
A 5 per cent event repeated an increasing number of times.
| Repetitions | P(at least once) |
|---|---|
| 1 tries | 5.00% |
| 5 tries | 22.6% |
| 14 tries | 51.2% |
| 25 tries | 72.3% |
| 50 tries | 92.3% |
| 100 tries | 99.4% |
Questions
What does independent mean?
That one event occurring does not change the probability of the other. Two coin flips are independent; drawing two cards without replacement is not, since the first draw changes what remains.
Why subtract the overlap for "or"?
Because adding the two probabilities counts the cases where both occur twice. Subtracting the overlap corrects it. This is the inclusion-exclusion principle.
How do I calculate at least one occurrence?
One minus the probability of no occurrences. If each trial has probability p of not happening, then n trials all failing is (1−p)ⁿ, and one minus that gives at least one success.
What if the events are mutually exclusive?
Then they cannot both occur, so P(A and B) is zero and P(A or B) is simply the sum. This calculator assumes independence rather than exclusivity, so it would need adjusting for that case.
How do probabilities relate to odds?
Odds compare favourable to unfavourable outcomes, while probability compares favourable to total. A probability of 25 per cent is odds of 1 to 3 against, or a decimal odds of 4.
For counting outcomes, see permutations and combinations. For repeated trials specifically, use binomial probability.