How to Calculate the Probability of Two Independent Events

To calculate the probability of two independent events, open the Probability Calculator, enter P(A) % and P(B) %, and read P(A and B) for the combined chance both happen, or the rows below it for "or," "neither," and "at least once in N trials."

What the Probability Calculator does

Two events are independent when one doesn't affect the other's odds — a coin flip and a dice roll, or rain tomorrow and your flight being delayed for an unrelated reason. For independent events, the Probability Calculator takes the individual chances of A and B and works out every combined probability people actually need: both happening, either happening, exactly one happening, neither happening, and the chance that an event with probability A happens at least once across a number of repeated trials.

How to calculate combined probability with Toolbench

1. Open the Probability Calculator.
2. Enter P(A) % — the probability of the first event, as a percentage (defaults to 50).
3. Enter P(B) % — the probability of the second, independent event (defaults to 30).
4. Enter Trials for "at least once" — how many times event A gets a chance to happen (defaults to 5).
5. Read P(A and B) in the main result box — the chance both A and B happen.
6. Check the rows below for P(A or B), P(A xor B), P(neither), P(not A), P(not B), and P(A at least once in N). Every field updates the results instantly as you type, and clicking any row copies its value.

If you enter a probability below 0% or above 100%, the calculator shows an error instead of a result, since a probability outside that range isn't meaningful.

A real worked example

With the default inputs — P(A) = 50%, P(B) = 30%, 5 trials:

Result	Value
P(A and B)	15%
P(A or B)	65%
P(A xor B)	50%
P(neither)	35%
P(not A)	50%
P(not B)	70%
P(A at least once in 5)	96.875%

P(A and B) is 50% × 30% = 15%. P(A or B) adds the two probabilities and subtracts their overlap (50% + 30% − 15% = 65%), so A and B aren't double-counted when both happen. P(A at least once in 5) is 1 minus the chance A never happens in any of 5 tries: 1 − 0.5⁵ = 1 − 0.03125 = 96.875%.

A second example with different odds

Say there's a 40% chance of rain tomorrow (event A) and, independently, a 20% chance your unrelated flight gets delayed (event B). Entering P(A) = 40, P(B) = 20, and 5 trials (five days of that same 40% rain chance) gives:

Result	Value
P(A and B)	8%
P(A or B)	52%
P(A xor B)	44%
P(neither)	48%
P(not A)	60%
P(not B)	80%
P(A at least once in 5)	92.224%

Even though any single day only has a 40% chance of rain, there's a 92.224% chance it rains on at least one of the five days — repeated trials compound quickly, which is exactly why "at least once" probabilities tend to feel higher than people expect.

What each result actually means

Result	Meaning
P(A and B)	Both A and B happen
P(A or B)	A happens, B happens, or both (inclusive or)
P(A xor B)	Exactly one of A or B happens, not both
P(neither)	Neither A nor B happens
P(not A)	A doesn't happen
P(not B)	B doesn't happen
P(A at least once in N)	A happens on at least one of N independent trials

Tips and common mistakes

• This only works for independent events. If knowing A happened changes the odds of B — like drawing two cards from the same deck without replacement — these formulas don't apply. The calculator assumes independence; it has no way to account for dependence between A and B.
• "Or" means inclusive or, not xor. P(A or B) counts the case where both happen. If you want exactly one but not both, use P(A xor B) instead.
• "At least once" uses A's probability, not B's. The trials field only repeats event A — set P(A) % to whichever event you're repeating before reading that row.
• Probabilities must stay between 0% and 100%. Entering a negative number or anything over 100 in either field produces an error rather than a result.
• Rounding is to 4 decimal places. Results like 96.875% are exact for the given inputs; trailing zeros are simply dropped, not rounded away.

Related tools

• Average Calculator — mean, median, mode, range and standard deviation of a list.
• Confidence Interval Calculator — turn a sample mean and standard deviation into a confidence interval.
• Percentage Calculator — percent of, percent change, increase and decrease.

Frequently asked questions

How do I find the probability that two independent events both happen?

Open the Probability Calculator, enter each event's probability in P(A) % and P(B) %, and read P(A and B) — it multiplies the two probabilities together, which is the rule for independent events.

What's the difference between P(A or B) and P(A xor B)?

P(A or B) is the chance that A happens, B happens, or both — it includes the overlap. P(A xor B) is the chance that exactly one of them happens, excluding the case where both do.

How does the "at least once in N trials" result work?

It takes event A's probability and calculates the chance A happens at least one time across N independent repeats, using 1 minus the probability A fails every single time: 1 − (1 − P(A))ⁿ.

Can I use this for events that aren't independent?

No. The formulas assume the two events don't influence each other's odds. If one event changes the probability of the other — like drawing cards without replacement — this calculator's results won't be accurate.

Is my data sent to a server?

No. The calculation runs in your browser as you type. If you're signed in, your inputs sync privately to your account; if you're signed out, nothing is saved anywhere.