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 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.
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.
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.
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.
One missing entry from the total of 1, and P(Y at least 3) by adding upwards.
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.
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.
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At the last stop, you learned 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?
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