Exercise 3: Round-Half-Up vs. Round-Half-Even in a Real Billing System — Possible Solution
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WHY IT'S A MEASURABLE MISTAKE, NOT A STYLE CHOICE
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This chapter's own verified 1,000-value experiment showed the exact
mechanism at play: rounding exactly-half amounts with round-half-up
produced a systematic bias of +500 out of a true total of 500,000
(a consistent 0.1% upward drift), purely because every tie broke the
same direction. Round-half-to-even, applied to the identical values,
produced zero bias, because the ties alternated between rounding up
and down and cancelled out.

A billing system processing many thousands of half-cent ties per day
is in exactly the same situation as this chapter's own experiment,
just at a larger and continuous scale. If it uses round-half-up, it
will systematically round every exact half-cent tie in the same
direction (up), which means the business consistently collects
slightly more than the mathematically exact total, every single day,
compounding over every transaction. This is not a random or
occasional error that averages out over time - it is a directional,
one-way bias with a predictable, nonzero size, exactly like the
verified +500 bias in this chapter's own 1,000-value case.

WHY THIS MATTERS BEYOND "IT'S A TINY AMOUNT"
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Even though any single half-cent rounding is a minuscule amount, the
verified experiment shows the bias doesn't shrink relative to volume
- it scales directly with the number of ties rounded, the same way
Chapter 1's own million-addition example showed error compounding
with the number of operations. At sufficient transaction volume, a
one-directional half-cent bias becomes a real, auditable, and
potentially legally significant discrepancy between the system's
reported totals and the mathematically correct ones - not because
any individual rounding was "wrong," but because the systematic
directional bias never has a chance to cancel itself out.

WHY THIS WORKS AS AN ANSWER
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The explanation applies this chapter's own verified 1,000-tie result
directly to the billing scenario rather than treating the two as
unrelated, correctly identifies that the problem is a directional
bias rather than ordinary random error, and explains why that bias
compounds with transaction volume instead of averaging away, tying
the conclusion back to a specific verified finding rather than
general intuition about rounding being "imprecise."