Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1 (or 0% to 100%). A probability of 0 means the event is impossible. A probability of 1 means the event is certain. A probability of 0.5 means the event is equally likely to happen or not happen. Probability is calculated as the number of favourable outcomes divided by the total number of possible outcomes, assuming all outcomes are equally likely.

How do you calculate nCr (combinations)?

nCr formula: n! / (r! x (n-r)!), where n is the total number of items and r is the number chosen. n! (n factorial) means n x (n-1) x (n-2) x ... x 1. Example: 8C3 = 8! / (3! x 5!) = (8 x 7 x 6) / (3 x 2 x 1) = 336 / 6 = 56. This means there are 56 ways to choose 3 items from 8 when order does not matter.

Probability Calculator

Q: What is the difference between combinations and permutations?

Combinations (nCr) count the number of ways to choose r items from n items when order does not matter. For example, choosing 3 people from a group of 10 for a committee: nCr = 10! / (3! x 7!) = 120 ways. Permutations (nPr) count the arrangements when order does matter. Choosing and arranging 3 people from 10 for first, second and third place: nPr = 10! / 7! = 720 ways. Permutations are always greater than or equal to combinations for the same n and r.

Q: What is a Z-score in normal distribution?

A Z-score measures how many standard deviations a value is from the mean of a normal distribution. Z = (X - mean) / standard deviation. A Z-score of 0 means the value equals the mean. A Z-score of 1 means the value is 1 standard deviation above the mean. In a standard normal distribution (mean=0, standard deviation=1), approximately 68% of values fall within 1 standard deviation, 95% within 2, and 99.7% within 3. Z-scores are used to find the probability that a value falls within a certain range.

Q: What is the binomial probability formula?

The binomial probability formula calculates P(X=k): the probability of exactly k successes in n independent trials, where each trial has probability p of success. Formula: P(X=k) = C(n,k) x p^k x (1-p)^(n-k). Example: The probability of getting exactly 3 heads in 10 coin flips (p=0.5): P(X=3) = C(10,3) x 0.5^3 x 0.5^7 = 120 x 0.125 x 0.0078125 = 0.1172 (approximately 11.7%).

Q: What is conditional probability?

Conditional probability P(A|B) is the probability that event A occurs given that event B has already occurred. Formula: P(A|B) = P(A and B) / P(B). For example, if P(rain) = 0.3 and P(cloudy and rain) = 0.25, then P(rain|cloudy) = 0.25 / P(cloudy). Conditional probability is the foundation of Bayes theorem and is used in medical diagnosis, spam filtering, and many real-world applications.

Q: How do you calculate dice probability?

For a fair die with 6 sides, each outcome has probability 1/6. The probability of rolling a specific number is 1/6. The probability of rolling at least a 4 (rolling 4, 5 or 6) is 3/6 = 1/2. For multiple dice, use the multiplication rule for independent events. The probability of rolling two sixes on two dice is (1/6) x (1/6) = 1/36. Use the basic probability calculator on this page and enter favourable outcomes and total outcomes to calculate any dice probability.

Q: What is the Poisson distribution?

The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate. The formula is P(X=k) = (lambda^k x e^-lambda) / k!, where lambda is the average number of events and k is the specific number you want the probability for. Common applications include: phone calls received per hour, defects per manufacturing batch, website visits per minute, and accidents per day on a road.

Key features

Everything the Probability Calculator Does

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Basic Probability P(A)

Calculate P(A), P(not A), P(A and B), P(A or B), and conditional probability P(A|B) from favourable outcomes or known probabilities.

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Combinations nCr Calculator

Calculates C(n,r) with full factorial step-by-step working shown. Handles n up to 170. Used for lottery odds, committee selection and choosing problems.

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Permutations nPr Calculator

Calculates P(n,r) with step-by-step working. Used for arrangement and ordering problems where sequence matters.

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Normal Distribution + Bell Curve

Enter mean, standard deviation and X to get Z-score, P(X<x) and P(X>x). For ranges, enter X2 for P(x1<X<x2). The unique shaded bell curve visualises the probability area.

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Binomial Probability Calculator

Given n trials, probability p and successes k: calculates P(X=k), P(X<=k) and P(X>=k) with full formula working. Plots a bar chart of the full distribution.

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Dice Probability Calculator

Roll any number of any-sided dice, see the result and probability. Enter a target sum to see the probability of rolling that total across multiple dice.

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Coin Flip Simulator

Flip any number of coins and see the results. Shows heads/tails counts, the actual ratio, and the probability of that exact outcome using the binomial formula.

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Step-by-Step Solutions

Every calculation shows the formula, substituted values and working line by line - ideal for students checking homework or learning the method.

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Instant Results

All calculations run instantly in your browser. No page reload, no server round-trip. Results appear immediately on clicking Calculate. No login required.

How to use

How to Use the Probability Calculator

Choose the right tab for your problem

Basic P(A) for simple probability and combined events. nCr/nPr for combinations and permutations. Normal Distribution for Z-score and bell curve problems. Binomial for repeated trials. Dice and Coin for simulations.

Basic probability: enter outcomes

Enter the number of favourable outcomes (A) and total possible outcomes (N). For example, rolling a 4 on a six-sided die: A=1, N=6. Optionally enter P(B) and P(A and B) for combined event calculations.

Combinations and permutations: enter n and r

Enter n (total items) and r (items chosen). The calculator shows both nCr and nPr with the full factorial calculation. For example, to find how many ways to choose 5 lottery numbers from 59: n=59, r=5.

Normal distribution: enter mean, SD and X

Enter the population mean, standard deviation and the value X you are interested in. The tool calculates the Z-score, P(X less than x) and P(X greater than x), and draws the bell curve with the relevant area shaded. For a range, also enter X2.

Binomial: enter n, p and k

Enter n (number of trials), p (probability of success per trial, between 0 and 1) and k (the specific number of successes you want the probability for). The tool shows P(X=k), P(X at most k) and P(X at least k) with the binomial formula working.

Read the step-by-step working

Every result panel shows the formula used, the values substituted in and each calculation step. Use this to verify your own working or to understand how the answer was reached before presenting it in an assignment or exam.

Competitor comparison

Probability Calculator: LazyTools vs Competitors

Most free probability calculators handle only one type of problem per page. LazyTools combines all five modes with step-by-step working and the unique visual bell curve in one tool.

Feature	LazyTools	Omnicalculator	Calculator.net	Symbolab
Basic probability	Yes	Yes	Yes	Yes
nCr / nPr with working	Yes	Yes	Yes	Yes
Normal distribution / Z-score	Yes	Yes	Yes	Yes
Bell curve visualisation	Yes (shaded)	Yes	No	No
Binomial distribution	Yes + bar chart	Yes	Yes	Yes
Dice probability calculator	Yes + simulator	Yes	Yes	No
Step-by-step working	Yes (all modes)	No	No	Yes (paid)
No login required	Yes	Yes	Yes	Limited free

Probability guide

Probability Formulas: Complete Reference

Probability is a branch of mathematics that quantifies uncertainty. Every probability value lies between 0 (impossible) and 1 (certain). This calculator covers five major areas of probability theory, each with their own formulas and applications.

Formula	Expression	Meaning
P(A)	Favourable / Total	Probability of event A
P(not A)	1 - P(A)	Probability A does not occur
P(A and B)	P(A) x P(B) if independent	Both A and B occur
P(A or B)	P(A) + P(B) - P(A and B)	At least one of A or B occurs
nCr	n! / (r! x (n-r)!)	Combinations: order does not matter
nPr	n! / (n-r)!	Permutations: order matters
Z-score	(X - mean) / SD	Standard deviations from mean
Binomial P(X=k)	C(n,k) x p^k x (1-p)^(n-k)	Exactly k successes in n trials
Poisson P(X=k)	(lambda^k x e^-lambda) / k!	k events given average rate lambda

Combinations nCr calculator: choosing without order

The combinations nCr calculator answers "how many ways can I choose r items from n, when order does not matter?" The formula is nCr = n! / (r! x (n-r)!). Factorials grow very quickly, so this calculator handles the arithmetic for you up to n=170. Common applications include lottery odds (choosing 6 from 49: C(49,6) = 13,983,816 combinations), selecting committee members from a group, choosing which questions to answer on an exam, and any selection problem where the arrangement of chosen items does not matter.

-- Example: UK National Lottery (6 from 59) nCr = n! / (r! x (n-r)!) = 59! / (6! x 53!) = (59 x 58 x 57 x 56 x 55 x 54) / (6 x 5 x 4 x 3 x 2 x 1) = 45,057,474 (about 1 in 45 million)

Normal distribution Z-score calculator

The normal distribution Z-score calculator converts a raw value X to a Z-score and then finds the corresponding probability. The Z-score formula is Z = (X - mean) / standard deviation. Once you have the Z-score, the cumulative distribution function (CDF) gives P(X less than x), the probability of observing a value below X. This is used in: quality control (what percentage of products fall outside tolerance?), exam marking (what percentage of students scored below 70?), medical statistics (is this patient's measurement within the normal range?), and finance (what is the probability of a loss exceeding this threshold?).

Binomial probability calculator step by step

The binomial probability calculator step by step shows the full working for P(X=k) in n independent trials with success probability p. The formula is P(X=k) = C(n,k) x p^k x (1-p)^(n-k). This applies whenever: there is a fixed number of trials, each trial is independent, each trial has the same two outcomes (success/failure), and each trial has the same probability p. Examples include: coin flips (p=0.5), quality control sampling (p=defect rate), medical trials (p=treatment success rate), and exam multiple choice (p=probability of guessing correctly).

Conditional probability calculator

Conditional probability P(A|B) is the probability of A occurring given that B has already happened. The formula is P(A|B) = P(A and B) / P(B). In the Basic P(A) tab, enter P(B) and P(A and B) to see the conditional probability. This is the foundation of Bayes theorem: P(A|B) = P(B|A) x P(A) / P(B), which updates the probability of A given new evidence B. Real-world applications include medical diagnosis (probability of disease given a positive test result), spam filtering and machine learning classification.

Poisson distribution calculator

The Poisson distribution calculator models the probability of k events occurring in a fixed interval when the average rate is known. The formula is P(X=k) = (lambda^k x e^-lambda) / k!, where lambda is the average number of events and e is Euler's number (approximately 2.71828). Poisson is used for: website visits per minute, phone calls per hour, typos per page, traffic accidents per day, and any scenario where events are rare, independent and occur at a constant average rate. The Poisson distribution is a limiting case of the binomial distribution when n is large and p is small, with lambda = n x p.

Dice probability calculator free

For a standard fair die, each face has equal probability 1/6. For a target sum across multiple dice, the number of ways to achieve that sum determines the probability. For two six-sided dice, the sum 7 has the most combinations (1+6, 2+5, 3+4, 4+3, 5+2, 6+1 = 6 ways out of 36), giving probability 6/36 = 1/6. The dice simulator tab rolls the dice and calculates both the simulated result and the theoretical probability of your target sum.

FAQ

Frequently Asked Questions

What is probability?▼

Probability measures how likely an event is, expressed between 0 (impossible) and 1 (certain). It is calculated as favourable outcomes divided by total possible outcomes, assuming all outcomes are equally likely. A probability of 0.5 means the event is equally likely to happen or not.

What is the difference between combinations and permutations?▼

Combinations (nCr) count selections where order does not matter. Permutations (nPr) count arrangements where order matters. Choosing 3 from 10 for a committee: C(10,3) = 120. Arranging 3 from 10 for 1st, 2nd, 3rd: P(10,3) = 720. Permutations are always larger than or equal to combinations.

How do you calculate nCr?▼

nCr = n! / (r! x (n-r)!). Example: 8C3 = 8! / (3! x 5!) = (8 x 7 x 6) / (3 x 2 x 1) = 336 / 6 = 56. Enter n and r in the nCr/nPr tab above for the full step-by-step working.

What is a Z-score in normal distribution?▼

A Z-score measures standard deviations from the mean: Z = (X - mean) / SD. Z=0 means the value equals the mean. Z=1 means 1 standard deviation above. About 68% of values fall within Z = -1 to +1, 95% within Z = -2 to +2, and 99.7% within Z = -3 to +3.

What is the binomial probability formula?▼

P(X=k) = C(n,k) x p^k x (1-p)^(n-k). Example: probability of 3 heads in 10 coin flips: P(X=3) = C(10,3) x 0.5^3 x 0.5^7 = 120 x 0.125 x 0.0078 = 0.117 (11.7%).

What is conditional probability?▼

P(A|B) = P(A and B) / P(B). It is the probability of A given that B has occurred. Example: P(rain|cloudy) = P(rain and cloudy) / P(cloudy). This is the basis of Bayes theorem and is used in medical diagnosis, spam filtering and machine learning.

How do you calculate dice probability?▼

For a fair d6, each face has probability 1/6. Rolling at least 4: probability = 3/6 = 0.5. Two dice summing to 7: 6 combinations out of 36 total = 1/6. Use the Dice tab above for any die configuration with an optional target sum probability.

What is the Poisson distribution?▼

Poisson models the probability of k events in a fixed interval given average rate lambda. P(X=k) = (lambda^k x e^-lambda) / k!. Used for phone calls per hour, defects per batch, website visits per minute. It is a limiting case of binomial with large n and small p where lambda = n x p.

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