# AND Probability Calculator

URL: https://speedycalculator.com/calculators/statistics/and-probability/
Last updated: 2026-10-30

> The AND probability calculator gives the chance that event A and event B both happen – by multiplying for independent events, using P(B | A) for dependent ones, or zero for mutually exclusive events – and shows the parts of A and B that do not overlap on a Venn diagram.

## Inputs

- Probability of A – P(A) (%)
- Probability of B – P(B) (%)
- How are A and B related?
- Probability of B given A – P(B | A) (%)

## Results

- P(A and B)
- Venn diagram
- A happens, B does not
- B happens, A does not
- P(A or B) for comparison
- P(A | B)
- P(B | A)
- Independent?

## Formula

```
P(A and B) = P(A) × P(B) (independent)
P(A and B) = P(A) × P(B | A) (dependent)
P(A and B) = 0 (mutually exclusive)
```

## Explanation

The word "and" in a probability question asks for the intersection: both things have to happen. Rolling a six and flipping heads, a train being late on Monday and on Tuesday, a part passing two separate inspections – each is an AND question. The answer is always at most the smaller of the two probabilities, because requiring a second event can only make success rarer.

The multiplication rule does the work. For independent events the probabilities simply multiply: a six (1/6) and heads (1/2) together have probability 1/12 = 8.33%. When the first event changes the odds of the second, multiply by the conditional probability instead – drawing a heart and then another heart without replacement is 13/52 × 12/51 = 5.88%. If the events can never happen together, the AND probability is zero.

Pick the relationship, enter P(A) and P(B) (and P(B | A) when needed) and read off P(A and B), the shaded overlap in the Venn diagram, the "A only" and "B only" parts and both conditional probabilities. A consistency check warns you when the numbers cannot all be true.

### How to use the AND Probability Calculator

- **Probability of A – P(A)**
- **Probability of B – P(B)**
- **How are A and B related?**
- **Probability of B given A – P(B | A)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The overlap is P(A)·P(B), P(A)·P(B | A) or 0 depending on the relationship chosen. Subtracting it from each probability gives the "only" regions; dividing it by P(A) or P(B) gives the conditional probabilities.

### Worked example

A shop finds that 60% of visitors browse the sale section (A) and 40% sign up for the newsletter (B). If the two are independent, P(both) = 0.6 × 0.4 = 24%; 36% browse but do not sign up and 16% sign up without browsing. If instead half of the browsers sign up, P(both) = 0.6 × 0.5 = 30%, and P(browsed | signed up) = 0.30 ÷ 0.40 = 75%.

### Everyday AND examples

QuestionCalculationAnswerTwo heads in two flips½ × ½25%Six on two dice1/6 × 1/62.78%Two hearts without replacement13/52 × 12/515.88%Two aces with replacement4/52 × 4/520.59%

## Worked example

Inputs — Probability of A – P(A): 60 %; Probability of B – P(B): 40 %; How are A and B related?: Independent.

Results — **P(A and B) = 24 %**; A happens, B does not = 36 %; B happens, A does not = 16 %; P(A or B) for comparison = 76 %; P(A | B) = 60 %; P(B | A) = 40 %.

## Key terms

- **Multiplication rule**: P(A and B) = P(A) × P(B | A).
- **Intersection**: The outcomes common to A and B.
- **Conditional probability**: P(B | A), the chance of B once A is known to have happened.

## FAQ

**What does "and" mean in probability?**
Both events must occur – the intersection A ∩ B.

**How do I calculate an AND probability?**
For independent events multiply: 60% × 40% = 24%. For dependent events use P(A) × P(B | A): 60% × 50% = 30%.

**What is the probability of rolling a six and flipping heads?**
1/6 × 1/2 = 1/12 ≈ 8.33%.

**Can P(A and B) be bigger than P(A)?**
Never. It cannot exceed the smaller of the two probabilities.

**What is P(A and B) for mutually exclusive events?**
Zero – they cannot happen together.

**How is AND different from OR?**
AND needs both events and multiplies; OR needs at least one and adds, subtracting the overlap.


## Sources

- [OpenStax – Introductory Statistics 2e, Chapter 3: Probability topics](https://openstax.org/books/introductory-statistics-2e/pages/3-introduction)
- [Khan Academy – Statistics and probability](https://www.khanacademy.org/math/statistics-probability)