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Two dice: totals are not equally likely

Part A · L02 · 5–10 minutes · Prerequisite: L01

The situation and question

Our generic task rule adds two independent six-sided dice. Before deciding whether a task succeeds, what does its unmodified total look like? This is a building block, not a claim about a particular Traveller edition.

Predict

Are a total of 2 and a total of 7 equally likely? Both are possible, but how many pairs of faces make each one?

Build

The + sign adds the results of independent dice. Each d(6) represents a fresh die. In tabletop notation this is 2d6. We use explicit addition here so the script clearly reports a total rather than a collection of individual faces.

output("Two independent d6 added", d(6) + d(6))

Two independent d6 added · DieRoll · mean 7.000

Run and read

There are 6 × 6 = 36 equally likely ordered pairs. A 2 needs (1, 1), so it has probability 1/36. A 7 comes from (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), or (6, 1): probability 6/36. The report's triangular shape reflects those counts; it is not a normal distribution.

The range is 2–12 and the mean is 7. Unlike the previous lesson's 3.5, this mean is also a possible result. That coincidence is not a requirement for an average.

Change one thing

Replace only the second die with d(4). Will the maximum rise or fall? Check: the new range is 2–10 and the mean is 3.5 + 2.5 = 6.

Try it yourself

Report the total of two four-sided dice. Explain why its middle total is most likely. Answer check: d(4) + d(4) ranges from 2 to 8; 5 has four of the sixteen pairs, so its probability is 1/4 and the mean is 5.

What you now know

Equally likely faces do not imply equally likely totals. Count the underlying pairs when checking a small model.

Where next

Previous: one die · Next: modifiers. The ability-score recipe later asks what changes when you discard a die instead of adding all of them.

Content ID: L02 · Review status: pilot