Probability Calculator

What this calculator does

Probability measures how likely an event is on a scale from 0 to 1, where 0 means impossible and 1 means certain. This calculator handles two common situations. The simple mode divides the number of favorable outcomes by the total number of equally likely outcomes and also reports the complement. The two-event mode takes the probabilities of two independent events and derives their intersections, union, exclusive outcomes, and the chance that neither occurs.

For example, if two independent quality checks each pass with probability 0.9, both pass with probability 0.81 and at least one passes with probability 0.99.

How to use this calculator

1. Choose the calculation type.
2. For a simple probability, enter the number of favorable outcomes and the total number of outcomes.
3. For two independent events, enter P(A) and P(B) as decimals or fractions between 0 and 1.
4. Click Calculate.

The headline shows the main probability. The rows below show every derived probability, including complements, the intersection, the union, and outcomes where exactly one event occurs.

Formulas

Simple probability: P(A) = favorable outcomes / total outcomes
Complement: P(A') = 1 − P(A)

Intersection: P(A ∩ B) = P(A) × P(B)          (independent events)
Union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Exclusive or: P(A Δ B) = P(A) + P(B) − 2 × P(A ∩ B)
Neither: P((A ∪ B)') = 1 − P(A ∪ B)
A only: P(A) × (1 − P(B))
B only: P(B) × (1 − P(A))

The union formula subtracts the intersection to avoid double-counting. The exclusive-or formula leaves only outcomes where exactly one event occurs.

Default example

The default two-event inputs are P(A) = 0.5 and P(B) = 0.4. Their complements are 0.5 and 0.6. The intersection is 0.5 × 0.4 = 0.2, and the union is 0.5 + 0.4 − 0.2 = 0.7. The chance that exactly one event occurs is 0.5, the chance that neither occurs is 0.3, and the individual exclusive outcomes are 0.3 for A only and 0.2 for B only.

In the simple mode, the default values of 3 favorable outcomes out of 8 total outcomes produce a probability of 0.375, a complement of 0.625, and a percentage of 37.5%.

Assumptions and limitations

• Probabilities must lie between 0 and 1 inclusive. Values outside that range are rejected.
• The two-event formulas assume that events A and B are independent. If one event changes the chance of the other, the multiplication rule for the intersection does not apply.
• The simple mode assumes equally likely outcomes.
• Mutually exclusive events are a special case: their intersection is zero, so their union equals the sum of their probabilities. The calculator does not detect mutual exclusivity automatically.
• Conditional probability, dependent events, and sampling without replacement are outside the scope of this page.
• Very small probabilities are displayed with limited decimal places and may appear as zero.

Frequently asked questions

What is the difference between the union and the intersection?

The intersection is the event that both A and B occur. The union is the event that at least one of them occurs, including the case where both occur. The union is always at least as large as either individual probability, while the intersection is never larger than the smaller of the two.

What does independence mean?

Two events are independent when the occurrence of one does not change the probability of the other. Coin flips and separate rolls of a die are independent. Drawing cards without replacement from a deck is not independent, because each draw changes the composition of the deck.

Can the union of two probabilities exceed 1?

Not for valid probabilities. The union formula subtracts the intersection, which keeps the result at or below 1. If the two events are mutually exclusive, the intersection is zero and the union is the sum, which is at most 1 as long as the individual probabilities are valid.

Source

Reference: Calculator.net — Probability Calculator.