--- 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: 2026-11-12 --- > **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 pays on the total of two fair dice rolled independently. Each pair has chance 1/6 × 1/6 = 1/36. A random variable is a rule assigning a number to each outcome. Here X adds the two faces. The rule is defined before rolling; the roll reveals its value. For equally likely outcomes, divide the count giving a value by the total count. For known unequal outcome probabilities, add the probabilities of outcomes giving that value. *(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.)* Count the pairs giving each total. Only double one gives 2, so P(X = 2) = 1/36. Four pairs give 5: 1 and 4, 2 and 3, 3 and 2, 4 and 1, so P(X = 5) = 4/36. Listing each total from 2 to 12 with its chance gives the probability distribution of X. These probabilities describe the model. Observed relative frequencies are counts divided by the number of rolls. In a finite run they may be close to the model or far from it; exact agreement is possible. *(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 the probabilities add to 36 / 36 = 1. Every roll produces exactly one total, so the eleven totals cover every roll with no overlaps. Every distribution totals 1 and has no negative entries, so you can find a missing probability. **Example.** A spinner stall has a random variable Y with values 1, 2, 3 and 4. Its probabilities are 0.2, 0.35, a smudge and 0.15. They total 1, so the smudge is 1 - 0.2 - 0.35 - 0.15 = 0.3. P(Y ≥ 3) means 3 or 4. One spin gives one score, so these events cannot both happen and their joint chance is 0. Add their probabilities: 0.3 + 0.15 = 0.45. At most 2 means adding the entries for 1 and 2 instead. X is a rule, not a number. It assigns a number to each outcome. Its distribution lists every possible value with its probability, so it answers questions about the total. Unlisted values have probability zero; for two dice, 13 cannot occur. The table keeps values and probabilities, not the faces that produced a total of 6. Different games can share the same distribution. **Recap.** A random variable turns an outcome into a number. Its distribution lists each value with its chance, built by counting. The chances total 1, which finds a missing one. For P(X ≥ k), add upwards. *Original lesson, question and flashcard revisions authored for this dataset.* ## 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)