Exercise 3: Why the Independence Assumption Was a Separate, Real Error — Possible Solution
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Squaring a single-event probability to get the combined probability of
two events is only mathematically valid if the two events are
genuinely statistically independent - meaning the first death occurring
has no effect at all on the real probability of the second death
occurring. Squaring assumes the two SIDS deaths were as unrelated to
each other as two separate coin flips would be.

That assumption would need to be true for the calculation to be valid:
nothing about being part of the same family - shared genetics, a
shared home environment, a shared exposure to any real risk factor for
SIDS - could make a second death more or less likely once a first had
already occurred within that specific family.

This is a real, separate error from the prosecutor's fallacy itself.
The prosecutor's fallacy is about which question a probability actually
answers (evidence-given-innocence versus innocence-given-evidence). The
independence assumption is a separate, earlier mathematical error in
how the 1-in-73-million number itself was calculated in the first
place - even before getting to which question it was being used to
answer. In reality, real medical research suggests a family that has
already experienced one SIDS death has a somewhat higher, not equal or
lower, real risk of a second one, since shared genetic or
environmental risk factors don't reset between children in the same
family - making Meadow's squared figure wrong on its own terms, before
the prosecutor's fallacy is even applied on top of it.

ANSWER: For squaring to be mathematically valid, the two deaths would
need to be genuinely statistically independent - meaning nothing about
being in the same family (shared genetics, shared environment) could
make a second SIDS death more likely once a first had occurred. This
is a real, separate error from the prosecutor's fallacy: it's a flaw in
how the underlying probability was calculated, distinct from the
separate error of which question that probability was then used to
answer.

WHY THIS WORKS AS AN ANSWER
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This correctly identifies the independence assumption as a distinct,
earlier mathematical error separate from the prosecutor's fallacy, and
explains precisely what condition would need to hold for the original
calculation to have been valid.