A Probability Calculator helps you quickly compute the likelihood of an event, from simple single-event odds to more advanced scenarios like combinations, permutations, conditional probability, and binomial trials. Whether you’re studying statistics, checking lottery-style odds, or validating A/B test results, this page explains the key concepts and how to use the calculator effectively.
Probability measures how likely an event is to occur, expressed between 0 (impossible) and 1 (certain), or as a percentage 0%–100%.
P( ext{event}) = frac{ ext{number of favorable outcomes}}{ ext{total possible outcomes}}
P( ext{event}) = frac{ ext{times event occurred}}{ ext{number of trials}}
Our Probability Calculator is designed to cover the most common needs:
For equally likely outcomes (e.g., rolling a fair die), use:
P(A) = frac{ ext{favorable}}{ ext{total}}
Sometimes it’s easier to compute the opposite:
P(A) = 1 - P(A^c)
If events are independent, multiply probabilities:
P(A cap B) = P(A) imes P(B)
When one event affects the other:
P(Amid B) = frac{P(A cap B)}{P(B)}
P( ext{2 red}) = frac{26}{52} imes frac{25}{51}
When order doesn’t matter (combinations) vs does (permutations):
{}_nC_r = frac{n!}{r!(n-r)!},qquad
{}_nP_r = frac{n!}{(n-r)!}
For independent trials with success probability :
P(X=k) = {n choose k} p^k (1-p)^{n-k}
Sum single binomial probabilities to get “at most/at least” results:
P(X le k)=sum_{i=0}^{k}{n choose i}p^i(1-p)^{n-i}
For large , a binomial can be approximated by a normal distribution with
mu=np,quad sigma=sqrt{np(1-p)}
Example 1: At least one success
A startup has a 20% chance () of a user upgrading on any visit. In 5 visits, the chance of at least one upgrade:
P(Xge1)=1-P(X=0)=1-(1-0.2)^5=1-0.8^5approx 0.6723
Example 2: Drawing balls without replacement
An urn has 5 blue and 7 red balls. What’s the probability of drawing 2 blue balls without replacement?
P=frac{5}{12} imesfrac{4}{11}=frac{20}{132}approx 0.1515
Example 3: Team selection
How many 4-person teams from 12 players?
{}_{12}C_4=frac{12!}{4!,8!}=495
Combinations count selections where order doesn’t matter (e.g., picking a committee). Permutations count arrangements where order matters (e.g., ranking winners 1st/2nd/3rd).
Use it when you have n independent trials, each with the same success probability , and you want the chance of exactly successes (or a cumulative range like at most/at least ).
Apply the complement rule: . Compute as “all failures,” which is often much simpler.
Independent events don’t affect each other’s probabilities (like separate coin flips). Dependent events change the probabilities as you go (like drawing cards without replacement).
Yes. Use single-event for basic odds, combinations/permutations for counting outcomes, and binomial for repeated trials (e.g., number of heads in many flips).
When is large and isn’t too small (a common rule of thumb: both and ≥ 10). It speeds up cumulative probability calculations.
Yes. If events cannot happen together, then . If they can overlap, subtract the intersection: .
Enter your numbers in the Probability Calculator on this page to compute exact or cumulative probabilities, combinations, permutations, and more—in seconds. Perfect for homework, planning experiments, gaming odds, and everyday decision-making.