Probability Calculator

What this tool does

Probability Calculator covers four common probability questions: a single event’s probability from favourable and total outcomes, the probability of two independent events both happening (AND), the probability of either happening (OR, correctly adjusted for mutual exclusivity), and the complement of an event. Each result shows its working formula.

Probability questions in a statistics class usually fall into a small number of recurring patterns: a basic favourable-over-total ratio, combining two events with AND or OR, or finding the complement of an event. Getting the right formula for each pattern, especially remembering when to subtract the overlap in an OR calculation, is where most mistakes happen.

This calculator walks through four of those patterns directly: a single event’s probability from a count of favourable and total outcomes, the AND probability of two independent events (multiplying), the OR probability of two events (adding, then subtracting the overlap unless they are mutually exclusive), and the complement of a single event (one minus its probability). Every result shows the exact formula used so the working is visible, not just the answer.

How to use Probability Calculator

1. Choose a Calculation: single event, P(A and B), P(A or B), or P(not A).
2. For a single event, enter Favourable outcomes and Total outcomes.
3. For combined events, enter P(A) and, where relevant, P(B) as decimals between 0 and 1.
4. Toggle A and B are mutually exclusive on if the two events cannot both happen in the same trial.
5. Read the resulting probability as a percentage, decimal and, for single events, as odds.
6. Check the notes for the exact formula applied to your inputs.

Tips for better results

• The AND and OR formulas for combined events here assume independence; if one event’s outcome affects the other’s probability, a conditional probability approach is needed instead, which this tool does not cover.
• Mutually exclusive events always have P(A and B) = 0, since they cannot both occur in the same trial by definition, which simplifies the OR formula to simple addition.
• The complement rule, P(not A) = 1 − P(A), holds unconditionally regardless of independence, since an event and its opposite always cover every possible outcome exactly once.
• Odds and probability are related but distinct: odds of 1:5 against correspond to a probability of 1/6, not 1/5.

Common use cases

• Checking a basic probability homework answer, such as the chance of drawing a specific card or rolling a specific die result.
• Working out the combined probability of two independent events both happening, such as two separate coin flips landing heads.
• Finding the probability that at least one of two events happens using the OR rule.
• Teaching the distinction between independent, mutually exclusive and complementary events with worked examples.

Common mistakes to avoid

• Using the independent-events AND formula (multiplying probabilities) for events that are not actually independent, which gives a wrong answer whenever one event influences the other.
• Forgetting to subtract the overlap term when calculating P(A or B) for events that are not mutually exclusive, which double-counts outcomes where both A and B happen.
• Confusing mutually exclusive with independent; two mutually exclusive events are never independent unless one of them has probability zero, since knowing one happened tells you the other definitely did not.