How to use the Probability Calculator
- Enter the probability of event A and event B as percentages.
- Enter the number of tries if you want the chance of at least one success for A.
- Read the combined probabilities.
- Remember the events must be independent: one must not change the chance of the other.
Formula used
For independent events: P(A and B) = P(A) × P(B). P(A or B) = P(A) + P(B) − P(A and B). P(not A) = 1 − P(A). P(at least one A in n tries) = 1 − (1 − P(A))ⁿ.
Worked example
Here is the Probability Calculator worked through with the example values shown above, so you can check each step by hand.
| Inputs | |
|---|---|
| Probability of event A | 30 % |
| Probability of event B | 50 % |
| Number of tries for event A | 5 |
| Result | |
| A and B both happen | 15% |
| A or B or both | 65% |
| Neither happens | 35% |
| Exactly one happens | 50% |
| A does not happen | 70% |
| A happens at least once in n tries | 83.19% |
The rules for independent events
If events are independent, P(A and B) = P(A) × P(B). With A at 30% and B at 50%, both happen 15% of the time. P(A or B) = P(A) + P(B) − P(A and B) = 65%, and neither happens 35% of the time.
Exactly one of them happens 50% of the time, which is the "either" figure minus the "both" figure. The chance that A does not happen is 1 − 0.30 = 70%.
At least once in several tries
The chance of at least one success is easiest to find from the opposite: the chance of never succeeding is (1 − p)^n, so at least once is 1 − (1 − p)^n. With p = 30% and 5 tries that is 83.19%.
For a coin (50%) in 3 flips the chance of at least one head is 87.5%. For a fair die (one chance in six) rolled six times it is only 66.5%, not 100%, which surprises many people.
Independent or not
Two events are independent when knowing the result of one does not change the chance of the other, such as separate coin tosses. Drawing cards from a deck without putting them back is not independent: after one ace is drawn, fewer remain.
For events that cannot happen together (mutually exclusive), such as rolling a 1 and a 2 on one die, P(A and B) is 0 and P(A or B) is the sum. This calculator assumes independence, so use the permutation and combination calculator for card and lottery problems.
Reading probabilities sensibly
A 5% chance per try is not 5% overall. Over 20 tries the chance of at least one hit is about 64%. This is why small risks that repeat, such as daily exposures, add up.
Probabilities describe long-run frequencies and do not make a single outcome certain. Past results do not change the odds of independent future events, which is the point behind the gambler’s fallacy.
Frequently asked questions
What is the probability of A and B?
For independent events, multiply the two probabilities.
What is the probability of A or B?
Add the probabilities and subtract the probability of both, to avoid counting that twice.
How do I find the chance of at least one success?
Calculate 1 − (1 − p)^n, where p is the chance per try and n the number of tries.
What does independent mean?
One event has no effect on the chance of the other.
Can I enter a decimal probability?
Enter it as a percentage, so 0.25 becomes 25.
Why is the chance of exactly one lower than either?
"Either" includes the cases where both happen. "Exactly one" excludes them.
This calculator gives estimates for information only. Actual figures from a bank, employer, tax authority or doctor may differ. It is not financial, tax, legal or medical advice.