Probability Calculator
Calculate probability easily with different methods
What Is the Probability Calculator?
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). This calculator supports basic probability plus three ways of combining probabilities: the complement rule, the AND rule for two independent events both occurring, and the OR rule for at least one of two events occurring.
Formal probability theory dates to correspondence between Blaise Pascal and Pierre de Fermat in the 1650s, sparked by a gambler's question about how to fairly split stakes in an interrupted game of chance — their exchange laid the groundwork for treating uncertainty mathematically rather than just intuitively. The basic favorable-over-total definition used here (sometimes called classical probability) is the simplest of several formal definitions of probability that later mathematicians developed, and it's still the most intuitive starting point for equally-likely outcomes like dice, coins, and cards.
For counting the number of favorable or total outcomes in more complex scenarios — like how many 5-card hands are possible from a 52-card deck — see the Combination Calculator. For summarizing a data set of observed outcomes rather than a theoretical probability, the Standard Deviation Calculator is a related but different statistics tool.
Probability Calculator Formula
- Basic: P = favorable outcomes / total outcomes
- Complement: P(not A) = 1 − P(A)
- AND (independent events): P(A ∩ B) = P(A) × P(B)
- OR: P(A ∪ B) = P(A) + P(B) − P(A) × P(B)
How Is the Probability Calculator Calculated?
Basic probability is simply the fraction of outcomes that count as a "win." The complement rule works because an event either happens or it doesn't, so the two probabilities must add to 1. The OR formula subtracts P(A) × P(B) to avoid double-counting outcomes where both events occur together.
That last point trips people up the most: if you simply added P(A) + P(B) without subtracting the overlap, any outcome where both A and B happen would get counted twice, inflating the result. Picture two overlapping circles in a Venn diagram — the overlapping middle region represents P(A ∩ B), and it's only counted once in the true union, so it has to be subtracted back out after adding the two full circles.
Probability Calculator Example
Rolling a standard six-sided die, the probability of rolling a 6 is 1/6 ≈ 0.1667 (1 favorable outcome out of 6 total).
AND: flipping a fair coin and getting heads (P = 0.5) while separately rolling a die and getting a 6 (P ≈ 0.1667) — since the two events are independent, P(both) = 0.5 × 0.1667 ≈ 0.0833 (about 8.3%).
OR: if P(A) = 0.3 and P(B) = 0.4 for two independent events, P(A ∪ B) = 0.3 + 0.4 − (0.3 × 0.4) = 0.7 − 0.12 = 0.58 (58%).
How to Use the Probability Calculator
Step 1
Select the type of probability calculation you need.
Step 2
Enter the required values (e.g. favorable/total outcomes, or P(A) and P(B)).
Step 3
Click Calculate to see the probability.
Step 4
Confirm your inputs are decimals between 0 and 1 (or a favorable/total count), not percentages.
Step 5
Switch between modes to explore how the same two events combine differently under AND versus OR.
Benefits
- Packages your finished probability into a branded image card that's easier to share in a group chat than a bare decimal.
- Covers four common probability scenarios in one calculator.
- Shows the exact formula used alongside the result.
- Handles independent-event AND/OR calculations without manual formula lookup.
- Removes the risk of forgetting to subtract the overlap term in the OR calculation.
- Free, instant, and requires no spreadsheet or textbook lookup.
Common Probability Calculator Scenarios
Scenario 1
Coin flip, dice roll, and card draw probability problems.
Scenario 2
Estimating the likelihood of two independent events both occurring.
Scenario 3
Homework or exam preparation involving basic probability theory.
Scenario 4
Quick sanity checks on probability reasoning before applying it to a larger statistics or data science problem.
Scenario 5
Board game or casino game odds questions involving independent trials.
Scenario 6
Comparing "at least one of these happens" scenarios using the OR rule.
Understanding Your Result
The result is a number between 0 and 1 (or as a percentage, between 0% and 100%) representing how likely the event is. A result near 0 means the event is unlikely; near 1 means it's nearly certain.
For the AND and OR modes specifically, the result reflects how the two individual probabilities you entered combine. An AND result is always less than or equal to either individual probability, since requiring both events narrows the outcomes; an OR result is always greater than or equal to either individual probability, since it captures more possible outcomes (either event, or both).
Tips
- Convert your decimal result to a percentage by multiplying by 100 for easier interpretation.
- The AND formula only applies to independent events — for dependent events, you need conditional probability instead.
- Always check that your probability inputs are between 0 and 1 before combining them.
- Use the OR result as a sanity check: it should never be smaller than either individual probability you entered.
- For "at least one success in several tries" problems, it's often easier to compute 1 minus the complement of all-failures, rather than the OR rule directly.
Common Mistakes
- Using the AND/OR formulas for dependent events, where the simple multiplication rule doesn't apply.
- Forgetting to subtract the overlap term in the OR formula, leading to double-counting.
- Entering probabilities as percentages (like 50) instead of decimals (like 0.5).
- Confusing "independent" events (one doesn't affect the other) with "mutually exclusive" events (they can't both happen) — these are different concepts with different formulas.
- Applying the basic favorable/total formula to outcomes that aren't actually equally likely.
Frequently Asked Questions
What is a simple example of basic probability?
Tossing a fair coin gives P(Heads) = 1/2 = 0.5, and rolling a fair six-sided die gives P(rolling a 6) = 1/6 ≈ 0.1667 — both are classic favorable-over-total examples.
When should I use the AND formula?
Use P(A ∩ B) = P(A) × P(B) when two events are independent — meaning one event happening doesn't affect the probability of the other, like separate coin flips or separate dice rolls.
What does the complement rule mean?
The complement rule states that the probability of an event NOT happening is 1 minus the probability that it does happen, since exactly one of the two outcomes (it happens, or it doesn't) must occur.
What does P(A ∪ B) mean?
It represents the probability that event A, event B, or both occur — the union of the two events, calculated by adding the individual probabilities and subtracting their overlap.
Can probability be greater than 1?
No, probability is always between 0 (impossible) and 1 (certain) — a result outside that range usually indicates invalid input values, like entering a probability greater than 1.
What does it mean if a calculated probability equals exactly 1?
A probability of 1 means the event is certain to happen — for example, the probability of rolling a number between 1 and 6 on a standard six-sided die.
Are AND and OR probability different for dependent versus independent events?
Yes — this calculator's AND/OR formulas assume independent events (where one event's outcome doesn't affect the other); dependent events require conditional probability, P(A ∩ B) = P(A) × P(B|A), which uses a different, more general formula.
What's the difference between theoretical and experimental probability?
Theoretical probability is calculated from known possible outcomes (like this calculator does), while experimental probability is observed from actually repeating a trial many times and counting results — they tend to converge as the number of trials increases, a pattern known as the law of large numbers.
Can probability be exactly 0?
Yes — a probability of 0 means the event is impossible, such as rolling a 7 on a standard six-sided die.
Can I share my probability result as an image?
Yes — tap Share and, on supported devices, your result is shared as a branded image card, not just a text link.
What's the difference between 'independent' and 'mutually exclusive' events?
Independent events don't affect each other's probability (like two separate coin flips), while mutually exclusive events can't both happen at once (like rolling a 2 and a 5 on the same single die roll) — they're different properties, and mutually exclusive events are actually never independent unless one has zero probability.
Why is the AND probability always smaller than either individual probability?
Because requiring two events to both happen is always at least as restrictive as requiring just one — multiplying two numbers between 0 and 1 always produces a result no larger than either original number.
How do I calculate the probability of at least one success across several independent trials?
It's usually easier to use the complement: calculate the probability that every trial fails, then subtract that from 1 — for example, the probability of at least one 6 in three die rolls is 1 − (5/6)³ ≈ 0.4213, rather than trying to apply the OR rule directly across three events.
References
Important Information
Enter probabilities as decimals between 0 and 1, not percentages.
Last updated: July 25, 2026