Probability Calculator
Calculate the probability of two events with AND, OR and NOT, a series of repeated independent events, or the probability of a normal distribution falling between two bounds.
How to use this calculator
- Choose what you are calculating: two events (AND/OR/NOT), a series of repeated independent events, or the probability of a normal distribution falling between two bounds.
- For two events, enter each probability and whether they are independent or mutually exclusive.
- For a normal distribution, enter the mean and standard deviation, then either bound — tick "no bound" on either side to treat it as infinite.
How the calculation works
Independent: P(A∩B) = P(A)×P(B)
Exclusive: P(A∩B) = 0
Either way: P(A∪B) = P(A) + P(B) − P(A∩B)
Series: P = P(A)^timesA × P(B)^timesB
Normal: P(Lb < X < Rb) = Φ((Rb−μ)/σ) − Φ((Lb−μ)/σ)- P(A∩B)
- Probability of both A and B
- P(A∪B)
- Probability of A or B (or both)
- P(not A)
- 1 − P(A) — the complement rule
- Φ
- The standard normal cumulative distribution function
Independence and mutual exclusivity are opposite ends of a spectrum, not the same idea. Independent events can co-occur; mutually exclusive events by definition cannot.
The general OR formula P(A) + P(B) − P(A∩B) works for any two events. It reduces to a plain sum only when P(A∩B) is zero, which is exactly the mutually-exclusive case.
For a series of independent repeats, probabilities multiply — the same reasoning as "independent" above, just applied more than once.
Worked example
Two independent events at 40% and 25%
- P(A and B) = 0.40 × 0.25 = 0.10, or 10%.
- P(A or B) = 0.40 + 0.25 − 0.10 = 0.55, or 55%.
- P(neither) = 1 − 0.55 = 0.45, or 45%.
Result: P(A and B) = 10%, P(A or B) = 55%
A normal distribution — mean 100, σ 15, between 85 and 115
- 85 and 115 are each exactly 1 standard deviation from the mean of 100.
- Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827.
- This is the well-known "about 68% within 1 standard deviation" figure.
Result: 68.27%
What probability measures
Probability puts a number on how likely something is, on a scale from 0 (impossible) to 1 (certain) — or, equivalently, 0% to 100%. It is a formal way of describing uncertainty: grounded either in counting equally likely outcomes, like a fair coin or die, or in observed long-run frequency, like the historical chance of rain under similar conditions.
Every probability calculation in practice reduces to combining a handful of individual probabilities using a small set of rules — for two events happening together, for either one happening, or for a whole run of repeated, independent events.
Independent, mutually exclusive, and why they are not opposites
Two ideas get confused constantly, and they sit at close to opposite ends of a spectrum. Independent events do not affect one another’s odds — flipping one coin tells you nothing about the next flip, so their probabilities can be multiplied directly. Mutually exclusive events cannot both happen at once — a single coin flip cannot land on both heads and tails, so there is no overlap to account for.
The distinction changes the arithmetic. For independent events, P(A and B) is a genuine multiplication, P(A) × P(B). For mutually exclusive events, P(A and B) is always zero, and P(A or B) collapses to a plain sum. Treating independent events as mutually exclusive, or the reverse, produces a wrong answer even when every input probability is correct.
Where probability shows up
The same handful of rules underpins fields that otherwise look nothing alike.
- Insurance and actuarial work — pricing a policy starts with the probability of the event it covers.
- Weather forecasting — a "30% chance of rain" is a probability estimate built from historical and modeled conditions.
- Genetics — predicting the odds of inheriting a particular trait from known parental genotypes.
- Quality control — the probability that a randomly sampled unit is defective, used to decide whether to reject a whole batch.
- Games and gambling — from card odds to dice, probability is the mathematical basis for how games are priced and played fairly.
Mistakes that trip people up
The gambler’s fallacy is the best-known trap — believing that after a run of coin flips landing heads, tails is somehow "due." For genuinely independent events, the odds reset completely on every trial; the coin has no memory of what came before.
A second common error is assuming events are independent when they are not, or the reverse. Drawing two cards from a deck without replacement is not independent — the odds on the second draw depend on what the first card was — which needs a different, conditional calculation than the "series" and "two events" modes here cover.
What this assumes, and where it stops
Assumptions
- Two-event mode assumes exactly two events, with the relationship between them stated explicitly rather than inferred. Normal-distribution mode assumes the underlying data is genuinely normally distributed.
Limitations
- Two-event mode handles only two events. More than two requires extending the same inclusion-exclusion logic, which grows more complex with each additional event.
- Does not handle conditional probability (events that influence but do not determine each other) — that needs Bayes’ theorem.
- Normal-distribution probabilities use the same erf approximation as the Z-Score Calculator, accurate to about 1.5 × 10⁻⁷ — far tighter than any practical use needs.
Common questions
What is the difference between independent and mutually exclusive?
Independent events do not affect one another’s probability — rolling two dice, the first roll tells you nothing about the second. Mutually exclusive events cannot both happen — a single coin flip cannot be both heads and tails. Independent events routinely co-occur; mutually exclusive events by definition never do.
Why subtract P(A and B) when calculating P(A or B)?
Because simply adding P(A) and P(B) counts the overlap twice — once within each probability. Subtracting P(A and B) removes that double-count. For mutually exclusive events there is no overlap to remove, which is why their OR probability is a plain sum.
How is the "series of events" mode different from AND?
AND (in two-event mode) asks for the probability of two different, once-off events both happening — like it raining today AND you missing a bus. Series mode asks for the probability of the same event happening repeatedly — like flipping heads 3 times in a row. Mathematically both multiply probabilities together; series mode is just built for the repeated-trial framing, with a repeat count instead of typing the same probability in twice.
Formula and content last reviewed on .
Built and maintained by Dev Mokshrajsinh.
Results are estimates for information only, not professional advice.
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