# 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.

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

## 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](01-one-die.html) · [Next: modifiers](03-modifiers.html).
The [ability-score recipe](../cookbook/ability-scores-4d6dl1.html) later asks
what changes when you discard a die instead of adding all of them.