Probability Calculator

What is a probability calculator?

A probability calculator takes the chances of two separate events and works out the combined odds of how they can play out together. Enter the probability of event A and the probability of event B as percentages, and the tool instantly reports four numbers: the chance that both happen, the chance that at least one happens, the chance that neither happens, and the chance that A does not happen.

This calculator treats the two events as independent — meaning the outcome of one has no effect on the outcome of the other, like two separate coin flips or two unrelated machines running at once. Everything runs in your browser, nothing is stored, and every answer comes straight from the standard rules of probability.

How does the probability calculator work?

Internally the tool converts each percentage to a decimal (50% becomes 0.5), applies the four formulas below, then converts the results back to percentages. With P(A) and P(B) as the two probabilities:

• Both events — P(A and B): multiply them. P(A and B) = P(A) × P(B). This is the multiplication rule for independent events.
• At least one event — P(A or B): add them, then subtract the overlap so it is not double-counted. P(A or B) = P(A) + P(B) − P(A) × P(B). This is the addition rule (inclusion–exclusion).
• Neither event — P(neither): multiply the chance each one fails to happen. P(neither) = (1 − P(A)) × (1 − P(B)).
• Not A — P(not A): the complement of A alone. P(not A) = 1 − P(A).

A few terms: a probability is always a number between 0 (impossible) and 1 (certain), shown here as 0% to 100%. The expression 1 − P(A) is the complement of A — the chance A does not occur. Notice that P(A or B) and P(neither) are themselves complements: at least one event happening is exactly the opposite of neither happening, so the two always add up to 100%.

Examples

Each example below matches the calculator exactly. Enter the same two percentages in the tool above and you will get the same four results.

Example 1: the verified 50% and 25% case

For P(A) = 50% and P(B) = 25% (so 0.5 and 0.25):

• A and B = 0.5 × 0.25 = 0.125 = 12.5%
• A or B = 0.5 + 0.25 − 0.125 = 0.625 = 62.5%
• Neither = (1 − 0.5) × (1 − 0.25) = 0.5 × 0.75 = 0.375 = 37.5%
• Not A = 1 − 0.5 = 0.5 = 50%

Note how 62.5% (A or B) and 37.5% (neither) add up to exactly 100% — they are complements of each other.

Example 2: two coin flips at 50% and 50%

For P(A) = 50% and P(B) = 50%:

• A and B = 0.5 × 0.5 = 25%
• A or B = 0.5 + 0.5 − 0.25 = 75%
• Neither = 0.5 × 0.5 = 25%
• Not A = 1 − 0.5 = 50%

This is the classic “two heads” question: there is a 25% chance both flips land heads and a 75% chance at least one does.

Example 3: a long shot and a near-certainty, 10% and 90%

For P(A) = 10% and P(B) = 90%:

• A and B = 0.1 × 0.9 = 0.09 = 9%
• A or B = 0.1 + 0.9 − 0.09 = 0.91 = 91%
• Neither = (1 − 0.1) × (1 − 0.9) = 0.9 × 0.1 = 0.09 = 9%
• Not A = 1 − 0.1 = 0.9 = 90%

Even though B is very likely, the chance that both happen is held down to 9% by the unlikely event A.

Quick reference table

This table shows each output for a few common input pairs, so you can sanity-check the tool at a glance (all values are percentages).

P(A)	P(B)	A and B	A or B	Neither	Not A
50%	25%	12.5%	62.5%	37.5%	50%
50%	50%	25%	75%	25%	50%
10%	90%	9%	91%	9%	90%
100%	0%	0%	100%	0%	0%
30%	30%	9%	51%	49%	70%

In every row, the “A or B” and “Neither” columns add to 100%, a handy way to confirm a result is consistent.

Common uses

• Coin flips and dice. Find the odds of two independent rolls or flips both landing a certain way, or of getting at least one success.
• Reliability and uptime. If two independent servers each have a known chance of failing, P(neither fails) and P(at least one fails) describe redundancy.
• Marketing and conversions. Estimate the chance that a visitor completes both of two independent steps, or at least one of two offers.
• Games and gambling. Combine the chance of two separate favourable events in a game where the draws do not influence each other.
• Everyday what-ifs. Will it rain on either of two unrelated trips? Will at least one of two independent applications succeed?

Tips and common mistakes

• Only use this for independent events. The multiplication rule P(A) × P(B) is correct only when one event does not affect the other. If they are linked, you need conditional probability instead.
• Do not just add for “or”. P(A) + P(B) over-counts the overlap. Always subtract P(A) × P(B), which is why 50% + 50% gives 75% for “A or B”, not 100%.
• Enter percentages, not decimals. Type 25, not 0.25, into a percentage field; the tool handles the conversion to a decimal for you.
• Keep values between 0 and 100. A probability cannot be negative or exceed 100%, so inputs outside that range are not meaningful.
• Use complements to check your work. P(A or B) + P(neither) should equal 100%; if it does not, an input was mistyped.

Limitations and notes

These formulas assume the two events are independent and that each is a single probability, not a conditional or time-dependent one. They do not apply to mutually exclusive events (where both cannot happen at once, so P(A and B) = 0) or to dependent events (where one outcome shifts the other’s odds). The tool also handles exactly two events; chaining more requires applying the rules step by step. Results are mathematical outputs for the numbers you enter — real-world probabilities are only as reliable as the estimates you start with. This calculator is for educational and general analytical use and is not a substitute for professional statistical advice.

Related tools

Keep exploring the statistics category: summarise a data set with the mean, median, mode calculator, measure spread with the standard deviation calculator, or count arrangements with the permutation and combination calculator.