PROBABILITY DISTRIBUTIONS
Calculate probabilities for Normal, Binomial, and Poisson distributions. Enter the required values to visualize the distribution, calculate probabilities, and see the formulas used to obtain the result.
Shaded green = P(X ≤ x). Dashed line = x.
WORKED SOLUTION
z = (x − μ) / σ = (6,000 − 5,778) / 800 = 0.278
P(X ≤ 6,000) = Φ(z) = 0.609302
A probability distribution is a mathematical function that describes how the values of a random variable are spread across possible outcomes. It tells us which outcomes are likely and which are rare, making it one of the foundational concepts in statistics and data analysis.
There are two main families of distributions:
The cumulative distribution function (CDF) gives the probability that a random variable X is less than or equal to a given value x, written as P(X ≤ x). This calculator works with three of the most commonly taught distributions: Normal, Binomial, and Poisson.
The Normal distribution (also called the Gaussian distribution or "bell curve") is a continuous probability distribution that is symmetric about its mean. It is one of the most important distributions in statistics because many natural phenomena — including measurement errors, test scores, and astronomical observations — tend to follow it.
A z-score tells you how many standard deviations a value x is from the mean:
where z is the standard score, x is the value you are evaluating, μ is the mean, and σ is the standard deviation. A positive z-score means x is above the mean; a negative z-score means x is below it.
Once you have the z-score, the cumulative probability is P(X ≤ x) = Φ(z), where Φ is the standard normal CDF. The calculator computes this automatically. To find the probability that X falls between two values a and b, calculate P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a) = Φ(zb) − Φ(za).
In the Zul'Elkoen Universe, the surface temperatures of a class of stars are normally distributed with a mean μ = 5778 K and a standard deviation σ = 800 K. What is the probability that a randomly selected star has a surface temperature of 6000 K or below?
There is approximately a 60.9% chance that a randomly selected star from this class has a surface temperature of 6000 K or below.
The Binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is appropriate when:
where C(n, k) = n! / (k! · (n − k)!) is the binomial coefficient ("n choose k"). The mean of the distribution is np and the variance is np(1 − p).
A Zul'Elkoen survey mission examines n = 20 candidate star systems for habitable planets, where each system has an independent p = 0.3 probability of containing a habitable planet. What is the probability of finding exactly k = 6 habitable systems?
There is approximately a 19.2% chance of finding exactly 6 habitable systems out of 20 surveyed.
The Poisson distribution models the number of events that occur within a fixed interval of time or space, when events occur independently at a constant average rate. It is appropriate when:
where e ≈ 2.71828 is Euler's number and k! is the factorial of k.
In the Zul'Elkoen Universe, a deep-space observation network records an average of λ = 2 supernovae per galaxy per century. What is the probability of observing exactly k = 3 supernovae in a single galaxy during a century?
There is approximately an 18.0% chance of observing exactly 3 supernovae in a single galaxy during one century.
| Distribution | Best Used For | Main Inputs | Example |
|---|---|---|---|
| Normal | Continuous measurements, errors, and natural variation | μ, σ, x | Star surface temperatures |
| Binomial | Number of successes across a fixed number of independent trials | n, p, k | Habitable planet detection |
| Poisson | Number of events occurring during a specified interval | λ, k | Supernovae per galaxy per century |
The Normal, Binomial, and Poisson distributions each model a different kind of random process. Below are examples from the Zul’Elkoen universe that genuinely satisfy the assumptions of each distribution— along with notes on when a fictional scenario might look like a fit but actually violate those assumptions.
The surface temperature of a main-sequence star is a continuous measurement shaped by many independent factors—mass, age, metallicity, and rotation rate. Within a single spectral class, these factors combine through the Central Limit Theorem to produce an approximately bell-shaped distribution. A survey of G-class stars in the Lurran Consortium’s trade corridors might find temperatures centered near 5,778 K (Sol’s temperature) with a standard deviation of roughly 800 K.
Biological traits follow the same logic. The heights of adult Beylthians—a tall, sapient species—are continuous and result from many independent genetic and environmental contributions, producing a Normal curve around a species-typical mean.
Why it fits: The variable is continuous, unbounded on either side, and the variation arises from many small independent factors. These are the conditions under which the Normal model is valid.
When not to use it: A trait like “number of limbs” is discrete and bounded, so it is not Normal. Similarly, orbital eccentricity (always between 0 and 1 and often right-skewed) does not produce a symmetric bell curve and should not be modeled as Normal.
When the Lurran Archaeological Society surveys 20 star systems for habitable planets, each system is an independent trial with a binary outcome: a habitable world is present (success) or it is not (failure). If the underlying fraction of habitable systems in the region is roughly 30%, the number of discoveries across 20 surveys follows a Binomial distribution with n = 20 and p = 0.3.
KAVOE sentry networks offer another legitimate case. If a defensive line deploys 12 independent sentry drones, and each drone has a 40% chance of detecting an incoming threat during a given engagement window, the total number of detections follows Binomial(n = 12, p = 0.4).
Why it fits: There is a fixed number of independent trials, each with exactly two outcomes, and the probability of success is constant from trial to trial. These are the three core assumptions of the Binomial model.
When not to use it: If the detection probability changes between drones—for example, because later drones learn from earlier failures—the trials are no longer identically distributed and the Binomial model breaks down. Similarly, “number of pirate attacks in a year” has no fixed upper bound on trials, so it belongs to the Poisson family, not the Binomial.
A supernova is a rare, independent event. Within a single galaxy, they occur at an approximately constant average rate—roughly one to three per century for a Milky Way-sized galaxy. Because there is no fixed number of “trials” (stars can explode at any continuous instant), the count of supernovae per century follows a Poisson distribution with λ ≈ 2.
A KAVOE deep-space sensor array recording cosmic ray strikes provides another clean example. Each strike arrives independently of the last, at a constant average rate per hour, with no upper limit on how many can arrive in the interval. The number of detections per hour is Poisson-distributed.
On a terraformed colony world, major volcanic eruptions are rare, independent geological events occurring at a roughly constant background rate. Modeling eruptions per decade as Poisson(λ = 4) is scientifically sound as long as the tectonic regime is stable.
Why it fits: Events occur independently, at a constant average rate, over a continuous interval with no fixed upper bound on the count. These are the defining assumptions of the Poisson process.
When not to use it: If events cluster—for instance, a coordinated pirate raid that triggers copycat attacks in the same sector—the independence assumption is violated and a simple Poisson model will understate the probability of clustered outbreaks. In that case a compound or overdispersed model is more appropriate.
Normal — use when the variable is continuous and variation comes from many small independent factors (Central Limit Theorem).
Binomial — use when there is a fixed number of independent, identical trials each with two outcomes and a constant success probability.
Poisson — use when rare, independent events occur at a constant average rate over a continuous interval with no upper bound on the count.
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