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.
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.
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.
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.
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.
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.
Last stop the rule was that independent chances multiply. The dice stall runs on it: two fair dice, each roll independent, so 6 × 6 = 36 equally likely pairs of faces. The stall pays on the total of the two, not on which die showed what.
The stall calls the total X. What sort of thing is X before the dice are thrown?
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.