Probability Distribution Table Calculator

A probability distribution table is a structured way to describe all possible outcomes of a discrete random variable X and their associated probabilities. Every row lists one value the random variable can take — written as P(X = x) — alongside how likely that outcome is.

Discrete distributions appear everywhere: the number of goals scored in a football match, the outcome of rolling a die, the number of defective items in a batch of 20, or the number of customers who arrive at a checkout in a minute. Whenever the outcomes are countable and finite, a probability distribution table gives you the full picture in one place.

From that table you can derive three key statistics: the expected value E(X) (the long-run average), the variance Var(X) (how spread out the values are), and the standard deviation SD(X) (variance in the same units as X). You also get the cumulative distribution function P(X ≤ x), which tells you the probability that X takes a value no greater than any threshold you choose.

Building a valid discrete probability distribution takes three steps:

1. List every possible outcome of the random variable in a row of values x₁, x₂, …, xₙ. These do not need to be integers or ordered, but ordering them makes the CDF column meaningful.
2. Assign a probability to each outcome. Each probability must be between 0 and 1 (inclusive). A probability of 0 means the outcome can never happen; 1 means it always happens.
3. Verify ΣP = 1. All probabilities must sum to exactly 1. If your total is 0.99 due to rounding, that is acceptable. If it is 0.9 or 1.1, your model is wrong.

Common sources of discrete distributions include: counting experiments (Binomial), waiting-time experiments (Geometric), rare-event counts (Poisson), and custom empirical data from observations.

The expected value E(X) is the long-run average of X over many repetitions. It answers the question: if you repeated this random experiment infinitely many times and averaged all the outcomes, what number would you approach?

Practical examples: If X is the payout on a lottery ticket where you win £10 with probability 0.05 and nothing with probability 0.95, then E(X) = 10(0.05) + 0(0.95) = £0.50. That is the fair ticket price — anything higher means the lottery profits; anything lower means the player has an edge.

Note that E(X) does not have to be a value X can actually take. The expected number of heads in three fair coin flips is 1.5 — an outcome that never occurs. E(X) is a summary statistic of the distribution, not a prediction of any individual outcome.

Variance Var(X) measures how spread out the values of X are around E(X). A high variance means outcomes vary widely; a low variance means most results cluster near the mean.

The two-step shortcut formula avoids computing squared deviations one by one:

Why the shortcut works: Expanding the definition Var(X) = E[(X − μ)²] algebraically gives Var(X) = E(X²) − [E(X)]². This is equivalent but much faster because you only need two sums over the table.

Standard deviation SD(X) is simply √Var(X). Because variance is in units of X², SD(X) brings the spread measure back to the same units as X itself, making it interpretable as "typical distance from the mean."

Quality control engineers use SD(X) to set tolerance bands. Finance analysts use Var(X) to compare investment risks. Teachers use SD(X) to understand how spread exam scores are around the class average.

The cumulative distribution function P(X ≤ x) gives the probability that the random variable takes a value no greater than x. It is computed by adding each probability from the first row down:

The CDF always starts at P(X ≤ x₁) = P(X = x₁) and ends at P(X ≤ xₙ) = 1. It is non-decreasing — each step adds a non-negative probability.

You can use the CDF column to answer questions like: "What is the probability that X is at most 3?" or "What is the probability that X is between 2 and 4?" (answer: CDF(4) − CDF(1)). These are the most common types of question in A-Level statistics exams.