Calculate odds, combinations, and permutations for any event. Results appear instantly as you type.
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Event Probability
Enter the probability of two events as decimals (0.25) or simple fractions (1/4).
Then choose whether the events are independent or dependent to see combined probabilities.
Event Type:
Independent Events: The outcome of Event A has absolutely no effect on
the outcome of Event B. Example: flipping a coin twice. Getting heads on the first flip
does not change your odds on the second flip.
Enter a decimal (e.g. 0.5) or a fraction (e.g. 1/2)
Enter a decimal (e.g. 0.33) or a fraction (e.g. 1/3)
Enter values above to see results.
Combinations (nCr) and Permutations (nPr)
Enter a total number of items (n) and the number of items being chosen or arranged (r).
Results for both combinations (order does not matter) and permutations (order matters)
appear side by side.
What is a Factorial? A factorial (written as n!) means multiplying a
whole number by every positive integer below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Factorials are the foundation of both combination and permutation formulas.
The size of your full set. Must be a whole number 0 or greater.
How many items are selected or arranged. Must be less than or equal to n.
Enter values above to see results.
Quick Scenarios: Dice and Coins
Select a scenario from the dropdowns and configure your outcome. The probability is
calculated using the binomial probability formula for repeated trials.
1. Choose Scenario
2. Number of Trials
Max 100 trials
3. Target Outcome
4. Die Face to Hit
Configure your scenario above.
Disclaimer: This calculator provides mathematical probabilities for
educational and general utility purposes. It is not intended for financial or gambling advice.
Probability calculations describe theoretical outcomes based on mathematical models and do not
guarantee real-world results.
The Ultimate Guide to Probability, Combinations, and Permutations
Whether you are a student tackling your first statistics class or a curious adult wanting to
sharpen your analytical thinking, understanding probability is one of the most useful skills
you can develop. The core concepts are simpler than they look. Here is a plain-language
breakdown of everything this calculator covers.
Probability is a number between 0 and 1 that describes how likely an event is to
occur. A probability of 0 means the event is impossible. A probability of 1 means
it is certain. Everything in between represents varying degrees of likelihood.
The basic formula is: P(Event) = Number of favorable outcomes divided by
the total number of possible outcomes. For example, a standard six-sided
die has 6 faces. The probability of rolling a 4 is 1/6, because there is one
favorable outcome (the face showing 4) out of 6 possible outcomes total.
Probability can be expressed in three equivalent formats. A fraction like 1/6 is
precise and exact. A decimal like 0.1667 is useful for quick comparison. A
percentage like 16.67% is the most intuitive format for everyday communication.
This calculator always shows all three at once so you can pick the format that
makes the most sense to you.
This is one of the most important distinctions in probability, and getting it wrong
leads to incorrect calculations.
Independent Events: Two events are independent when the result of
one event has zero effect on the probability of the other. Flipping a coin and
rolling a die at the same time is a classic example. Getting heads on the coin does
not change the odds of rolling a 6. The probability of both happening together is
simply P(A) multiplied by P(B). If P(coin = heads) = 1/2 and P(die = 6) = 1/6,
then P(both) = 1/2 x 1/6 = 1/12.
Dependent Events: Two events are dependent when the outcome of the
first event changes the probability of the second. Drawing cards from a deck without
replacing them is a textbook example. If you draw the Ace of Spades first, only 51
cards remain for the next draw, and the Ace of Spades can no longer appear again.
For dependent events, you need to know P(B given A has occurred), which is called
a conditional probability.
This calculator uses P(A and B) = P(A) x P(B|A) for dependent events. When you
select the "Dependent" toggle, it treats your P(B) input as the conditional
probability P(B given A already happened).
Both combinations and permutations deal with selecting items from a larger group,
but the critical difference is whether the order of selection matters.
Combinations (nCr) - Order Does NOT Matter: Use combinations when
you only care which items are selected, not the sequence. Imagine choosing 3 people
from a group of 10 to form a committee. The committee {Alice, Bob, Carol} is
identical to {Carol, Alice, Bob}. The formula is:
nCr = n! / (r! x (n-r)!)
Permutations (nPr) - Order DOES Matter: Use permutations when
the sequence of selection is important. Imagine assigning the same 3 people to
three specific officer roles: President, Vice-President, and Treasurer. Now
{Alice, Bob, Carol} is different from {Carol, Alice, Bob} because each person
fills a different role. The formula is:
nPr = n! / (n-r)!
Because permutations count every unique ordering, the permutation result is always
greater than or equal to the combination result for the same n and r. Specifically,
nPr = nCr x r!.
Two events are mutually exclusive when they cannot both happen at
the same time. On a single coin flip, you cannot get both heads and tails
simultaneously. These outcomes are mutually exclusive.
This matters most when calculating P(A or B): the probability that at least one of
two events occurs. There are two cases:
If A and B ARE mutually exclusive: P(A or B) = P(A) + P(B).
You simply add the probabilities because there is no overlap.
If A and B are NOT mutually exclusive: P(A or B) = P(A) + P(B)
- P(A and B). You must subtract P(A and B) to avoid counting the overlap
(cases where both A and B happen) twice.
This calculator uses the general formula (subtracting the overlap) because it
applies in all cases. If your events truly are mutually exclusive, P(A and B) will
be 0, and the formula simplifies automatically.
A factorial (written with an exclamation mark, like n!) is the
product of all positive integers from 1 up to n. So 4! = 4 x 3 x 2 x 1 = 24, and
6! = 720. Factorials grow extremely fast. 20! is already over 2.4 quintillion.
In probability, factorials appear in the combination and permutation formulas
because they count the total number of ways to arrange items. The problem is that
standard 64-bit computers have a maximum safe integer of 9,007,199,254,740,991.
Once a factorial exceeds this limit, standard integer arithmetic produces rounding
errors. 21! already exceeds this boundary.
This calculator handles large numbers by using JavaScript's built-in
BigInt type for exact integer arithmetic when n is large, and by
simplifying the combination formula before multiplying, which reduces the magnitude
of intermediate values. This approach lets the tool handle values up to n = 170
reliably. Beyond that, the numbers grow so astronomically large that they lose
practical meaning for most real-world calculations.