Math
Probability Calculator
Calculate basic, compound, Bayesian, and binomial probabilities
P(A)50.0000%
P(not A)50.0000%
Odds for1 : 1
Odds against1 : 1
## Probability Calculator โ Basic, Compound, Bayes & Binomial
Probability quantifies uncertainty. Our calculator covers the four most common probability scenarios: single-event probability with odds conversion, compound events (AND/OR), Bayesian updating, and binomial experiments.
### Basic Probability
For a single event A with probability P(A):
- **P(not A)** = 1 โ P(A) โ complement rule
- **Odds for** A = P(A) : P(not A)
- **Odds against** A = P(not A) : P(A)
### Compound Probability โ AND and OR
**P(A and B)** โ both events occur:
- Independent: P(A) ร P(B)
- Mutually exclusive: 0
- General: P(A) + P(B) โ P(A or B)
**P(A or B)** โ at least one event occurs:
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
### Conditional Probability
The probability of A given that B has already occurred:
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
This appears in many real-world situations: the probability of rain given cloudy skies, the probability of disease given a positive test, the probability of fraud given unusual spending.
### Bayes' Theorem
Update a belief when new evidence arrives:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B|A) \cdot P(A) + P(B|\neg A) \cdot P(\neg A)}$$
**Medical example**: A disease affects 1% of the population. A test is 99% accurate (1% false positives). If your test is positive, what is the actual probability you have the disease?
P(disease) = 0.01, P(positive|disease) = 0.99, P(positive|no disease) = 0.01
โ P(disease|positive) โ **50%** โ far less than 99%!
This counterintuitive result is explained by the low base rate of the disease.
### Binomial Probability
For n independent trials each with probability p of success:
$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$
Use this for: coin flips, quality control sampling, survey responses, or any yes/no experiment repeated a fixed number of times.
Frequently Asked Questions
What is the difference between P(A and B) and P(A or B)?
P(A and B) is the probability that both events occur simultaneously (intersection). P(A or B) is the probability that at least one of the events occurs (union). The addition rule gives: P(A or B) = P(A) + P(B) โ P(A and B). For mutually exclusive events where both cannot both occur at once, P(A and B) = 0.
What does "independent events" mean?
Two events are independent if the occurrence of one has no effect on the probability of the other. For independent events: P(A and B) = P(A) ร P(B). For example, two coin flips are independent โ the first flip has no effect on the second.
What are mutually exclusive events?
Mutually exclusive events cannot both occur at the same time. For example, a single coin flip cannot be both heads and tails. When events are mutually exclusive: P(A and B) = 0 and P(A or B) = P(A) + P(B).
What is conditional probability?
Conditional probability P(A|B) is the probability of event A occurring, given that event B has already occurred. The formula is: P(A|B) = P(A and B) / P(B). It reflects how knowledge of B changes the probability of A.
How does Bayes' theorem work?
Bayes' theorem updates a prior probability with new evidence. If you know P(A) (prior), P(B|A) (likelihood), and P(B|not A) (false positive rate), then P(A|B) = P(B|A) ร P(A) / P(B). It's used in medical testing: a positive test doesn't mean you have the disease โ the result depends on the test's accuracy and the disease's base rate.
What is binomial probability?
Binomial probability calculates the chance of exactly k successes in n independent trials, each with probability p of success: P(X=k) = C(n,k) ร pแต ร (1โp)โฟโปแต. For example, the probability of exactly 3 heads in 8 coin flips.