Compare exact, tail, and between-range probabilities for binomial, Poisson, normal, and uniform distributions from their defining parameters.
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Distribution
Fixed number of independent trials, each with the same success probability.
Query
P(X = 5)
0.25
For Binomial(n = 10, p = 0.5), the probability of exactly 5 successes is 24.6094%.
0.25
P(X = 5)
62.3%
P(X ≤ 5)
62.3%
P(X ≥ 5)
5
Mean
Mean
5
Variance
2.5
Std dev
1.58
Use exact mode for one outcome, cumulative modes for tails, and between mode for inclusive discrete ranges.
PMF applies to discrete distributions such as binomial and Poisson. PDF applies to continuous distributions
such as normal and uniform, where a single exact point has zero probability and the area under the curve
carries the probability.
Probability distribution calculator: compare exact, tail, and range probabilities
A probability distribution calculator helps you answer the practical question behind most homework, exam, and analytics workflows: what is the probability of an exact outcome, of being below or above a threshold, or of landing inside a range? This page covers binomial, Poisson, normal, and uniform distributions, showing PMF or PDF values, tail probabilities, interval probabilities, and the core summary statistics that define each model.
Which distribution should you use?
Choose the binomial distribution when you have a fixed number of independent trials and each trial has the same success probability. Typical examples include the number of heads in ten coin flips, pass counts in a short quiz, or the number of email opens in a fixed batch when each message is treated as the same Bernoulli trial.
Choose the Poisson distribution when you are counting events in a fixed interval and the average event rate is known or assumed stable. It is commonly used for arrivals per hour, defects per metre, or calls per minute when you need a count model rather than a continuous curve.
Choose the normal distribution when values cluster around a mean and you need a lower-tail, upper-tail, or between-bounds probability on a continuous scale. Exam scores, measurement error, and standardized z-score work are common examples.
Choose the uniform distribution when every value in a closed interval is equally likely. That model is simpler than the others: probability comes entirely from interval length, which makes it useful for random waiting windows, idealized random-number draws, and baseline teaching examples.
Binomial: P(X=k) = C(n,k) × p^k × (1−p)^(n−k)
Probability of exactly k successes in n trials.
Poisson: P(X=k) = e^(−λ) × λ^k / k!
Probability of exactly k events given average rate λ.
Normal CDF: Φ(x) = ½[1 + erf((x−μ) / (σ√2))]
Cumulative probability up to x for a normal distribution.
Uniform PDF = 1 / (b − a)
Constant density inside the interval [a, b] for the continuous uniform distribution.
Exact, tail, and between probabilities
Most users do not actually want a raw formula value; they want a probability statement. That is why this calculator separates exact or density mode from tail and interval modes. In discrete distributions, exact mode answers questions like P(X = 5). Tail modes answer questions such as P(X ≤ 5) or P(X ≥ 5), and between mode answers inclusive ranges like P(3 ≤ X ≤ 7).
For continuous distributions, an exact point has zero probability, so the PDF is density rather than probability. In practice, lower-tail, upper-tail, and between modes are more useful because they return areas under the curve. If you are solving a normal-distribution problem from a textbook, that area is usually the quantity the question is really asking for.
This distinction is one of the biggest sources of confusion in statistics classes. A PMF value for a discrete distribution is a genuine probability for one outcome. A PDF value for a continuous distribution is not. The calculator keeps both visible so you can see the difference rather than mixing the two ideas together.
How the supported distributions work
Binomial mode combines a fixed trial count n with a success probability p. The mean is n × p and the variance is n × p × (1 − p). That makes binomial distributions most useful when the number of opportunities is known in advance and each one is still active after previous outcomes.
Poisson mode uses the average event rate λ. In a Poisson process, the mean and variance are both λ, which is why the dispersion of the distribution grows as the average count increases. This is a natural model for independent counts in time or space when the rate is approximately constant over the interval you care about.
Normal mode is governed by the mean μ and standard deviation σ. The calculator also reports the z-score at the selected point because standardization is often the bridge between a raw-value question and a normal-probability lookup. Uniform mode is simpler: the density stays constant inside [a, b], so interval probabilities reduce to the fraction of the support interval covered by the selected range.
Worked examples and interpretation
Suppose a fair coin is flipped ten times and you want the probability of exactly five heads. Binomial mode with n = 10, p = 0.5, and exact k = 5 gives about 0.2461. If the question changes to at most five heads, the cumulative probability rises to about 0.6230 because you are now adding every probability from zero through five successes.
Now suppose calls arrive at an average rate of three per hour and you want the probability of at least two calls. Poisson mode with λ = 3 and the upper-tail query gives about 0.8009. That is a different question from the probability of exactly two calls, which is smaller because it refers to one count only.
For a standard normal distribution, the probability of lying between −1 and 1 is about 0.6827. That common benchmark is useful because it illustrates how much of a normal curve sits inside one standard deviation of the mean. For a Uniform(0, 10) distribution, the probability of being between 2 and 8 is exactly 0.6 because the covered interval is 6 units long out of a total support length of 10.
Frequently asked questions
What is the difference between PMF and PDF?
A probability mass function (PMF) applies to discrete distributions like binomial and Poisson, giving the exact probability of each outcome. A probability density function (PDF) applies to continuous distributions like normal and uniform — it gives density, not probability, for a single point. That is why this page exposes lower-tail, upper-tail, and between-range areas for continuous models instead of pretending that one exact point carries non-zero probability.
Why is the normal PDF value sometimes greater than 1?
The PDF gives probability density, not probability. The probability of any single exact value is zero for a continuous distribution. The PDF can exceed 1 as long as the total area under the curve equals 1. This often surprises people when the standard deviation is very small, because the curve becomes narrow and tall while the total area still remains exactly 1.
When should I use binomial instead of Poisson?
Use binomial when the number of trials is fixed and each trial has the same success probability. Use Poisson when you are counting how many events occur in an interval and the key parameter is the average rate λ rather than a fixed number of opportunities. In some large-n, small-p situations the Poisson distribution approximates the binomial distribution, but the modelling assumptions are still different.
What does the CDF tell me?
The cumulative distribution function (CDF) tells you the probability of being at or below a threshold. That makes it the natural building block for lower-tail questions and for between-range probabilities, which are found by subtracting one CDF value from another. If you want the probability of being above a threshold, the upper-tail probability is usually 1 minus the lower-tail probability at that threshold.
