Pick a calculator with the six buttons above the card: Discrete Uniform, Binomial, Geometric, Negative Binomial, Hypergeometric or Poisson; each keeps its own values. Learn more about switching calculators
Pick a Probability Type: the full distribution, or one of P(X = k), P(X < k), P(X ≤ k), P(X > k), P(X ≥ k); changing it clears the result and the k box. Learn more about the query types
Type the parameters of the chosen distribution, for example a and b for the uniform calculator or n and p for the binomial one; a red note flags a value of the wrong kind at once. Learn more about the uniform calculator
For a query other than the full distribution, type the value k (the uniform calculator calls it x) in the box that appears below the parameters. Learn more about binomial probabilities
Click Calculate; it stays grey until every box is valid. The query card shows the probability to six decimals, then the cards for the mean, the variance and the tool's own quantities. Learn more about the results
In the full-distribution mode, read the PMF chart and the value list under the cards; Distribution Info repeats the formula. Learn more about the Poisson calculator
Read the panel on the right: it explains the active calculator and, once you pick one, the chosen query type, with links to the sections below. Learn more about the geometric calculator
Click Reset to empty the boxes of the active tool and remove its result; the other tools are left as they were. Learn more about resetting
The row of buttons above the calculator holds the six tools of this page: Discrete Uniform, Binomial, Geometric, Negative Binomial, Hypergeometric and Poisson Distribution Calculator. The page opens on the uniform calculator; click a button to switch, and the blue heading of the card names the active tool. Each calculator keeps its own boxes, so the values you typed into one are not carried into another, and the result of the tool you leave is dropped when you come back to it.
Adding ?tab=2 to ?tab=6 after the page address opens a given calculator directly, in the order of the buttons; the buttons themselves do not change the address. Every tool has the same layout: the Probability Type box, its parameter boxes, the Calculate and Reset buttons, the result cards and, in the full-distribution mode, the PMF chart. The panel on the right of each tool explains the active calculator and the chosen query type. Which tool fits which situation is explained in Choosing the Right Discrete Distribution.
Selecting Probability Query Types
DemoThree query types on ten coin flips
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Each distribution calculator offers six query modes for flexible probability calculations. All values (full distribution) displays the complete probability mass function across all possible outcomes, showing the PMF chart and detailed probability table. This mode reveals the distribution's shape and identifies the most likely values.
P(X = k) calculates the exact probability of a specific outcome. For a binomial distribution with n=10 and p=0.5, P(X=5) gives the probability of exactly 5 successes. This mode uses the distribution's PMF formula directly.
P(X < k) computes cumulative probability for values strictly less than k, summing probabilities from the minimum up to k-1. P(X ≤ k) includes k itself in the sum. These queries answer questions like "What's the probability of fewer than 3 defects?" versus "At most 3 defects?"
P(X > k) and P(X ≥ k) calculate right-tail probabilities, useful for questions like "What's the probability of exceeding 10 events?" The greater-than query sums from k+1 onward, while greater-or-equal starts at k. All calculators display results prominently with the query notation and six-decimal precision.
Changing the type clears the result and empties the k box (the x box of the uniform calculator), so type the value again and click Calculate. The six types are compared in Understanding PMF vs CDF, and each has its own section, starting with The Full Distribution Query.
Using the Discrete Uniform Calculator
DemoA die: the full distribution, P(X ≤ 2), a value outside the range
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The discrete uniform distribution models scenarios where all outcomes have equal probability, like rolling a fair die or drawing a random integer. Enter a (minimum) as the smallest possible value and b (maximum) as the largest. For a standard die, set a=1 and b=6.
The calculator computes n = b - a + 1 (count of possible values) and probability = 1/n for each outcome. The PMF chart displays uniform-height bars showing equal probabilities. Mean equals (a+b)/2, the midpoint of the range. Variance follows the formula ((n²-1)/12).
Query specific values to see how uniform probability works. For a=1, b=6, P(X=3) = 1/6 ≈ 0.1667, identical to any other face. P(X ≤ 2) = 2/6 = 0.3333 since two values (1,2) fall in that range. The uniform distribution serves as a baseline for comparing more complex distributions and models perfectly random selection.
Both boxes take whole numbers only (a decimal is refused with "Must be an integer"), and b must exceed a, otherwise Calculate answers "b must be greater than a"; a query value outside a…b is refused with "x must be between a and b". The result cards show n, P(X=x), the mean and the variance; for a=1, b=6 they read 6, 0.1667, 3.5000 and 2.9167. The formulas are explained in The Discrete Uniform Distribution.
Calculating Binomial Distribution Probabilities
DemoTen coin flips: full distribution, P(X = 5), k above n
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The binomial distribution models the number of successes in a fixed number of independent trials, each with the same success probability. Enter n (number of trials) as the total independent repetitions and p (success probability) as the probability for each trial, between 0 and 1.
For coin flipping, n=10 flips with p=0.5 for a fair coin models total heads. For manufacturing quality control with 2% defect rate, use p=0.02 with n equal to sample size. The calculator displays the PMF chart showing the distribution's shape—symmetric when p=0.5, right-skewed when p<0.5, left-skewed when p>0.5.
Mean equals np (expected successes), variance equals np(1-p). For n=20, p=0.3, expect 6 successes on average with variance 4.2. The PMF uses the formula C(n,k) × p^k × (1-p)^(n-k) where C(n,k) is the binomial coefficient "n choose k."
Query P(X=k) for exact probabilities, P(X≤k) for cumulative "at most k successes." The binomial models coin flips, quality control sampling, survey responses, medical trials—any scenario with fixed trials and constant success probability.
The n box takes a positive whole number, the p box a decimal between 0 and 1 (digits and a point only), and k may not exceed n ("k cannot exceed n"). For n=10, p=0.5 the cards show q=0.5000, mean 5.0000, variance 2.5000 and standard deviation 1.5811; P(X=5)=0.246094 and P(X≤3)=0.171875. The formula is explained in The Binomial Distribution.
The geometric distribution models the number of trials needed until the first success occurs. Unlike binomial which fixes trial count, geometric asks "How many attempts until success?" Enter p (success probability) as the constant probability for each trial.
For p=0.2 (20% success rate), the calculator shows probabilities for needing 1, 2, 3... trials. P(X=1) = p (immediate success), P(X=2) = (1-p)×p (fail then succeed), and so on. The PMF chart displays exponentially decreasing bars—early success is most likely, long sequences less probable.
Mean equals 1/p, the expected number of trials. For p=0.2, expect 5 trials on average until first success. Variance equals (1-p)/p². Standard deviation measures spread around this expectation. The distribution has infinite support (theoretically could take forever), but the calculator displays up to reasonable maximum based on negligible probabilities.
Use geometric for questions like "How many job applications until an offer?" or "Attempts needed to roll a six?" The memoryless property means past failures don't affect future trial probabilities—each attempt has the same success chance regardless of history.
This calculator counts trials, so k starts at 1 ("Must be a positive integer"). For p=0.2 the cards show mean 5.0000 and variance 20.0000; P(X=5)=0.081920 and P(X>5)=0.327680, the chance that the first five trials all fail. The formula is explained in The Geometric Distribution.
Negative Binomial Distribution Calculator
Demor = 5, p = 0.3: full distribution, P(X = 10), P(X ≤ 10)
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The negative binomial distribution generalizes the geometric distribution, modeling failures before achieving a target number of successes. Enter r (number of successes) as the target success count and p (success probability) as the probability per trial.
For r=5, p=0.3, the calculator shows the distribution of failures before the 5th success. The random variable X represents failure count, not total trials. Mean equals r(1-p)/p, expected failures before r successes. For r=5, p=0.3, expect about 11.67 failures before accumulating 5 successes.
Variance equals r(1-p)/p², measuring variability in failure count. The PMF uses binomial coefficients: C(k+r-1, k) × p^r × (1-p)^k where k is the number of failures. The distribution resembles binomial but extends indefinitely since failure count is unbounded.
When r=1, negative binomial reduces to geometric distribution. Use this for scenarios like "How many defective items before finding 10 acceptable ones?" or "Customer rejections before closing 3 sales?" The calculator handles both specific failure counts P(X=k) and cumulative queries P(X≤k) for "at most k failures."
Here k counts failures, so it starts at 0 ("Must be a non-negative integer"). For r=5, p=0.3 the cards show mean 11.6667 and variance 38.8889; P(X=10)=0.068710 and P(X≤10)=0.484509. The formula is explained in The Negative Binomial Distribution.
Hypergeometric Distribution for Sampling Without Replacement
DemoA five-card hand: full distribution, P(X = 3), K above N
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The hypergeometric distribution models sampling without replacement from a finite population containing two types of items. Enter N (population size) as total items, K (success states) as items of the target type, and n (sample size) as the number drawn.
For drawing cards from a 52-card deck, N=52 total cards, K=13 hearts, n=5 cards drawn. The calculator computes probabilities for k hearts in the 5-card hand. Unlike binomial (which assumes replacement), hypergeometric accounts for changing probabilities as items are removed.
Mean equals nK/N, expected successes in the sample. For N=100 items with K=30 defective, drawing n=10 gives mean 3 defective items. Variance equals n×K×(N-K)×(N-n)/(N²×(N-1)), incorporating the finite population correction factor.
The PMF uses the formula: [C(K,k) × C(N-K, n-k)] / C(N,n) where C denotes combinations. Valid k values range from max(0, n-(N-K)) to min(n,K)—you can't draw more hearts than exist or more cards than sampled.
Use hypergeometric for quality control (sampling from production batch), lottery probabilities (drawing winning numbers), or ecological sampling (capturing tagged animals). The distribution approaches binomial as N→∞ with K/N=p constant.
The calculator refuses K>N ("K cannot exceed N") and n>N ("n cannot exceed N"), and a k outside the valid range is refused with "k must be between …". For N=52, K=13, n=5 the cards show mean 1.2500 and variance 0.8640, and P(X=3)=0.081543 is the chance of exactly three hearts in a five-card hand. The formula is explained in The Hypergeometric Distribution.
The Poisson distribution models the number of events occurring in a fixed interval when events happen independently at a constant average rate. Enter λ (lambda - average rate) as the expected event count per interval.
For λ=3.5 events per hour, the calculator shows probabilities for 0, 1, 2, 3... events in an hour. P(X=0) = e^(-λ) gives the probability of zero events. The PMF peak occurs near λ—the most likely outcome approximates the average rate.
Mean and variance both equal λ, a unique Poisson property. For λ=5, expect 5 events with variance 5 and standard deviation √5 ≈ 2.236. The distribution is right-skewed for small λ, approaches symmetry as λ increases. When λ>10, normal approximation becomes accurate.
The PMF formula: P(X=k) = (λ^k × e^(-λ)) / k! where k is the event count. The calculator sums probabilities for cumulative queries. P(X≤3) for λ=5 gives probability of 3 or fewer events, useful for capacity planning.
Apply Poisson to phone calls per hour, website visits per day, radioactive decay counts, rare disease cases, customer arrivals, or any scenario with independent events at constant rate. The distribution assumes infinite potential events in the interval with very small individual probabilities. For the formula worked out step by step, PMF and CDF charts, and all comparison types, see the Poisson distribution calculator.
The λ box takes any positive decimal. For λ=3.5 the cards show mean 3.5000 and variance 3.5000; P(X=5)=0.132169 and P(X≤3)=0.536633. The formula is explained in The Poisson Distribution.
Calculating, Reading the Results and Resetting
DemoA refused pair, the result cards, Reset
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The Calculate button stays grey until every required box holds a valid value: the parameters, and the k (or x) box for every query type except the full distribution. While you type, a red note under a box flags a value of the wrong kind at once, such as "Must be a positive integer" or "Must be between 0 and 1"; relations between boxes are checked when you click, for example "b must be greater than a", "k cannot exceed n" or "K cannot exceed N", and the result stays empty until the note is cleared.
A successful click fills the right-hand side. For a query type other than the full distribution, the first card shows the query in symbols, such as P(X=5), with its probability to six decimals. Below it the parameter cards follow: the mean μ and the variance σ2 in every tool, plus the tool's own quantities, n and P(X=x) for the uniform calculator, q=1−p and the standard deviation for the binomial one. Distribution Info repeats the PMF formula, and in the full-distribution mode the PMF chart and the value list follow. Reset empties every box of the active tool and removes its result; the query type and the other five tools are left as they were. For the die, a=1, b=6, the cards read n=6, P(X=x)=0.1667, mean 3.5000, variance 2.9167. What the mean and variance of a discrete distribution measure is explained in What Are Discrete Probability Distributions?.
What Are Discrete Probability Distributions?
A discrete probability distribution describes the probabilities of countable outcomes for a random variable. Unlike continuous distributions which assign probability to intervals, discrete distributions assign probability to specific values: 0 heads, 1 head, 2 heads, etc.
The probability mass function (PMF) gives P(X=k) for each possible value k. PMF values must be non-negative and sum to 1 across all possible outcomes. The PMF completely characterizes the distribution—knowing all individual probabilities determines every property.
Discrete distributions model counted phenomena: number of successes, defects, events, trials, or occurrences. The random variable takes integer values (sometimes all non-negative integers, sometimes a finite range). Examples include coin flips (0 to n heads), website visits (0 to infinity), cards drawn (0 to min(n,K) target cards).
Key properties include mean μ (expected value, center of distribution), variance σ² (spread around mean), and standard deviation σ (variance square root). The cumulative distribution function (CDF) gives P(X≤k), accumulating probabilities from minimum to k.
Selecting the appropriate distribution requires matching the scenario's structure to distribution assumptions. Discrete Uniform applies when all outcomes are equally likely with known minimum and maximum: dice rolls, random integer selection, lottery numbers.
Binomial suits fixed trial counts with constant success probability and independence: quality control samples, coin flipping, survey yes/no responses, multiple choice guessing. Each trial must be independent with identical success chance.
Geometric models waiting time until first success when trials are independent with constant probability: attempts until sale, rolls until six, iterations until convergence. Use when counting trials, not successes.
Negative Binomial generalizes geometric to multiple successes: failures before r-th success, defects before finding r acceptable items. Reduces to geometric when r=1. Appropriate when counting failures while waiting for fixed success count.
Hypergeometric handles sampling without replacement from finite populations with two categories: card hands, quality sampling from batches, capture-recapture methods. Critical when sample size is substantial relative to population—replacement assumption breaks down.
Poisson applies to rare events in fixed intervals when occurrences are independent with constant rate: phone calls per hour, website hits, accidents per month, rare disease cases. Works when individual event probability is tiny but opportunities are numerous. Approximates binomial when n is large, p is small, and np is moderate.
The discrete uniform distribution puts the same probability on each of the whole numbers from a to b: there are n=b−a+1 values and P(X=x)=n1 for every one of them, 0 elsewhere. A fair die is the standard case, a=1, b=6, P(X=x)=61.
The mean is the midpoint 2a+b and the variance is 12n2−1; for the die, 3.5 and 1235≈2.9167. Cumulative questions are counts: P(X≤x)=nx−a+1, so P(X≤2)=62 for the die. A common slip is to use b−a for the number of values; the count includes both ends, which is why n=b−a+1.
The binomial distribution counts the successes in n independent trials that each succeed with probability p. Its PMF is P(X=k)=(kn)pk(1−p)n−k for k=0,1,…,n: the binomial coefficient counts the ways to place k successes among n trials, and the powers give the probability of one such arrangement.
The mean is np and the variance np(1−p). For n=10 fair coin flips, p=0.5: the mean is 5, the variance 2.5, and P(X=5)=(510)/210=252/1024≈0.246094; the chance of at most 3 heads is (1+10+45+120)/1024≈0.171875. A common mistake is to apply the binomial when draws are made without replacement from a small population; then the trials are not independent and the hypergeometric distribution is the right model.
The geometric distribution counts the trials up to and including the first success, when every trial succeeds independently with probability p. Its PMF is P(X=k)=(1−p)k−1p for k=1,2,3,…: k−1 failures, then one success. The tail is simple, P(X>k)=(1−p)k, the chance that the first k trials all fail.
The mean is p1 and the variance p21−p. For p=0.2: mean 5, variance 20, P(X=5)=0.84⋅0.2=0.08192 and P(X>5)=0.85=0.32768. The distribution is memoryless: after any number of failures, the number of further trials needed has the same distribution as at the start. The common mistake is mixing the two conventions, trials until success (support from 1, used by this calculator) and failures before success (support from 0); the means differ by exactly 1.
The negative binomial distribution counts the failures that occur before the r-th success in independent trials with success probability p. Its PMF is P(X=k)=(kk+r−1)pr(1−p)k for k=0,1,2,…: the last trial is the r-th success, and the coefficient counts the arrangements of the other r−1 successes among the first k+r−1 trials.
The mean is pr(1−p) and the variance p2r(1−p). For r=5, p=0.3: mean 0.35⋅0.7≈11.6667, variance ≈38.8889, and P(X=10)=(1014)⋅0.35⋅0.710≈0.068710. With r=1 the distribution is the geometric distribution in its failures-before-success form. The common mistake is to read k as the total number of trials; the total is k+r.
The hypergeometric distribution counts the successes in a sample of n items drawn without replacement from a population of N items of which K are successes. Its PMF is P(X=k)=(nN)(kK)(n−kN−K) for k from max(0,n−(N−K)) to min(n,K): favourable hands over all hands.
The mean is NnK and the variance N2(N−1)nK(N−K)(N−n); the last factor, N−1N−n, is the finite population correction that makes the variance smaller than the binomial np(1−p). For a five-card hand from a deck, N=52, K=13 hearts, n=5: mean 1.25, variance ≈0.8640, and P(X=3)=(552)(313)(239)=2598960286⋅741≈0.081543. The common mistake is to use the binomial with p=K/N; that assumes replacement and overstates the spread.
The Poisson distribution counts events that occur independently at a constant average rate λ per interval. Its PMF is P(X=k)=k!λke−λ for k=0,1,2,…, and both the mean and the variance equal λ.
For λ=3.5 calls per hour: P(X=0)=e−3.5≈0.030197, P(X=5)=1203.55e−3.5≈0.132169, and the chance of at most 3 calls is e−3.5(1+3.5+6.125+7.1458)≈0.536633. The Poisson distribution approximates the binomial when n is large and p small with np=λ. The common mistake is to use a rate for the wrong interval: 3.5 per hour is 7 per two hours, so λ must be rescaled with the interval before k is queried.
The probability mass function (PMF) gives the probability of each exact value: P(X=k). For a die, PMF(3) = 1/6, the probability of rolling exactly 3. PMF bars in calculator charts show these individual probabilities. PMF values must be non-negative and sum to 1.
The cumulative distribution function (CDF) gives P(X≤k), the probability of at most k: all values from minimum through k. For a die, CDF(3) = 3/6 = 0.5, the probability of rolling 1, 2, or 3. CDF is the sum of PMF values up to k.
CDF increases from 0 to 1 as k increases. For discrete distributions, CDF is a step function—flat between integer values, jumping upward at each possible outcome by the PMF value. CDF(k) - CDF(k-1) equals PMF(k), relating the functions.
PMF answers "exactly k" questions while CDF answers "at most k" questions. Calculate P(X>k) as 1 - CDF(k). Calculate P(a < X ≤ b) as CDF(b) - CDF(a). The calculator query options implement these relationships: P(X<k) = CDF(k-1), P(X≤k) = CDF(k), P(X>k) = 1 - CDF(k), P(X≥k) = 1 - CDF(k-1).
The CDF visualizer tools display step functions showing cumulative probability accumulation, complementing the PMF bar charts in these calculators.
All values (full distribution) is the opening query type of every calculator. It needs only the parameters: Calculate then lists P(X=k) for every value of the support, up to a display cap for the distributions with an infinite support (the geometric, negative binomial and Poisson calculators stop a few standard deviations past the mean), draws the PMF chart and shows the mean and the variance. No k box is shown in this mode.
Use it first to see the shape of the distribution: the die gives six equal bars of 61, ten fair coin flips give a symmetric hump around 5, and λ=3.5 gives a right-skewed hump peaking at 3. The probabilities in the list sum to 1 (to rounding), which is the second axiom of a PMF. The PMF and its two rules are taught in The two fundamental axioms and the three ways to show it in Representations: the three views.
Exact Probability P(X = k)
P(X = k) gives the probability of one exact value, read straight from the PMF; the uniform calculator calls the value x. Type the parameters, choose this type, type k and click Calculate: the answer is shown with six decimals under its label, for example P(X=5)=0.246094 for ten fair coin flips, P(X=3)=0.166667 for a die, P(X=5)=0.132169 for λ=3.5.
The value must lie in the support: the binomial calculator refuses k>n, the uniform one refuses x outside a…b, and the geometric one needs k≥1. The usual mistake is to answer an "at most" question with an exact probability; "at most k" needs the sum in At Most: P(X ≤ k). Working with single PMF values is taught in Calculations using the PMF.
Strictly Less Than: P(X < k)
P(X < k) is the probability that X is strictly smaller than k: the sum of the PMF over all support values below k, so k itself is left out. For a die, P(X<3)=P(1)+P(2)=62≈0.333333; for the geometric distribution with p=0.2, P(X<5)=1−0.84=0.5904.
In CDF terms this is F(k−1), one step before the CDF value at k. The common mistake is to read "fewer than 3" as "3 or fewer"; the two differ by exactly P(X=k), which is why the calculator offers both this type and At Most: P(X ≤ k). The step shape of a discrete CDF, which makes this distinction matter, is taught in The CDF of a discrete variable.
At Most: P(X ≤ k)
P(X ≤ k) is the cumulative distribution function F(k): the sum of the PMF over every support value up to and including k. For a die, P(X≤2)=62≈0.333333; for ten fair coin flips, P(X≤3)=0.171875; for λ=3.5, P(X≤3)=0.536633.
This is the query behind "at most", "no more than" and "up to" questions, and the one whose values the CDF table of a textbook lists. The difference F(k)−F(k−1) recovers P(X=k), and 1−F(k) gives the right tail, so every other query type can be written with this one. The CDF is defined in The definition of the CDF and its use in Using the CDF.
Strictly Greater Than: P(X > k)
P(X > k) is the right tail without k: the probability of a value strictly above k, equal to 1−F(k). For the geometric distribution with p=0.2 it has a closed form, P(X>5)=0.85=0.32768, the chance that the first five trials all fail; for a die, P(X>3)=63.
For an infinite support the calculator sums the complement rather than the tail itself, which is also the quickest way by hand: compute F(k) and subtract from 1. The usual mistake is to answer "more than k" with 1−F(k−1), which includes k; that is the next type, At Least: P(X ≥ k). Complements and tails are covered in Calculations using the PMF and Connection to CDF.
At Least: P(X ≥ k)
P(X ≥ k) is the right tail including k: 1−F(k−1), the probability of k or more. For a die, P(X≥3)=64≈0.666667; for the geometric distribution with p=0.2, P(X≥5)=0.84=0.4096, the chance that the first four trials fail.
This is the query behind "at least", "no fewer than" and "k or more" questions, and the one used for a p-value style question such as "how likely are 8 or more heads in 10 flips", which gives 102445+10+1≈0.054688. The difference between this type and Strictly Greater Than: P(X > k) is exactly P(X=k). The relation to the CDF is taught in Connection to CDF and The CDF of a discrete variable.