--- title: "Discrete Random Variables and Probability Distributions" description: "A random variable attaches a number to each outcome, and its distribution lists every value the variable can take with the probability of taking it. Those probabilities have to total 1." canonical: https://lightmysky.com/learn/mathematics/discrete-random-variables-and-probability-distributions-mt_yKztNkvvq3 source: https://lightmysky.com/learn/mathematics/discrete-random-variables-and-probability-distributions-mt_yKztNkvvq3.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Discrete Random Variables and Probability Distributions A random variable attaches a number to each outcome, and its distribution lists every value the variable can take with the probability of taking it. Those probabilities have to total 1. Subject: Mathematics · Area: Probability · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/discrete-random-variables-and-probability-distributions-mt_yKztNkvvq3 ## Ready when they can - Write the distribution of the total when two dice are added - Use the total of 1 to find a missing probability in a table - Find P(X ≥ 3) from a given distribution ## Lesson: Every Value, and the Chance of Each The dice stall takes two dice and pays on the total. The two rolls are independent, and each fair die has six faces, so the product rule gives 6 × 6 = 36 equally likely pairs. The stall does not care which die showed what. It cares about one number per roll: the total. Something that turns each outcome into one number like that is called a random variable, and this one gets the letter X. Before the dice land X has no value. After they land it has exactly one. *(drawing: Thirty-six squares, one for each pair of faces, all equally likely. Every square carries a total, and several squares carry the same one.)* So work out how many squares carry each total. Only one pair adds to 2, the double one, so P(X = 2) = 1 / 36. Four pairs add to 5: 1 and 4, 2 and 3, 3 and 2, 4 and 1, so P(X = 5) = 4 / 36. Do that for every total from 2 to 12 and you have listed every value X can take alongside the chance of taking it. That list is the probability distribution of X, and it is the whole game written down. *(drawing: The number of pairs behind each total, climbing to six at a total of 7. Totals 10, 11 and 12 carry on down the other side at 3, 2 and 1.)* Add those counts: 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36, so they add to 36 / 36 = 1. That is not a coincidence. Every roll produces exactly one total, and the eleven totals cover every roll with no overlaps, the same complete-set fact as before. Any distribution totals 1 with no negative entry, which makes a missing one findable. **Example.** A spinner stall has a random variable Y with values 1, 2, 3 and 4. The card shows probabilities 0.2, 0.35, then a smudge, then 0.15. The four values cover every spin and cannot overlap, so they total 1, and the smudged entry is 1 - 0.2 - 0.35 - 0.15 = 0.3. Now the stall's real question: P(Y ≥ 3) means 3 or anything above, so add upwards: 0.3 + 0.15 = 0.45. At most 2 adds downwards instead. Keep the two ideas apart. X is not a number, it is the rule turning an outcome into a number, and the distribution is everything that rule can produce with the chance of each. With the distribution in hand you can forget the dice: every question about the total is answerable from that list. A value with no entry in the list has probability zero, which is why 13 never appears. **Recap.** A random variable is a rule attaching a number to each outcome, and its probability distribution lists every value it can take with the probability of taking it. Build one by counting how many equally likely outcomes give each value. The probabilities always total 1, because every outcome produces exactly one value, which is how a missing entry is found. For P(X ≥ some value), add the entries from there upwards, and downwards for at most. ## Practice 30 questions on this page, each with its working shown. ## Needs first - [Independent Events and the Multiplication Rule](https://lightmysky.com/learn/mathematics/independent-events-and-the-multiplication-rule-mt_MK8EPHEUdN) ## Opens up - [Joint, Marginal and Conditional Probability](https://lightmysky.com/learn/computing/joint-marginal-and-conditional-probability-mt__Rj-hOSMEw) - [Probability Spaces: Sample Spaces, Events and Axioms](https://lightmysky.com/learn/mathematics/probability-spaces-sample-spaces-events-and-axioms-mt_4gsBeqJtIQ) - [Microstates, Multiplicity and Boltzmann's Entropy](https://lightmysky.com/learn/science/microstates-multiplicity-and-boltzmanns-entropy-mt_55I3VvYMc6) - [Expected Value and Variance of a Discrete Random Variable](https://lightmysky.com/learn/mathematics/expected-value-and-variance-of-a-discrete-random-variable-mt_PFdo0Nm3FP)