Exercise 3: Why "$1,500,000 Over 1,000 Weeks" Makes Sense — Possible Solution
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THE APPARENT TENSION
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This chapter's own incident-cost example established E[X] = $1,500 per
week, yet also established that $1,500 is never actually a real
outcome for any single week (the only possible weekly costs are $0,
$500, $5,000, or $50,000). It might seem strange, then, to claim the
total cost over 1,000 weeks would be around $1,500,000.

WHY IT'S STILL A REASONABLE STATEMENT
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Per this chapter's own distinction, expected value is a LONG-RUN
AVERAGE, not a prediction of any single outcome. Over many repeated
weeks, some will cost $0 (about 70% of them), some $500, some $5,000,
and a rare few $50,000 - and as the number of weeks grows very large,
the actual total cost divided by the number of weeks gets closer and
closer to the expected value, $1,500 per week, even though no
individual week ever lands on that number. Multiplying $1,500 by 1,000
weeks is exactly applying that same long-run-average reasoning to a
longer time horizon: it's not claiming every week (or even any week)
costs $1,500 - it's claiming that across enough weeks, the highs
(rare $50,000 catastrophes) and lows (frequent $0 weeks) balance out
to that average.

WHY THIS IS USEFUL FOR BUDGETING DESPITE THE VARIANCE
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This chapter's own worked example also showed a large standard
deviation (~$7,057) relative to the mean, meaning any SINGLE week's
actual cost could differ wildly from $1,500. But that same variability
tends to average out over a long enough total (1,000 weeks), which is
exactly why "expected total cost" is a genuinely useful planning
number for a reserve fund sized over a long period, even though it
would be a poor way to predict any one specific week's bill.

WHY THIS WORKS AS AN ANSWER
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The explanation is grounded directly in this chapter's own "long-run
average, not a single prediction" distinction, extending it from a
single week to a thousand-week total, and explicitly reconciles it
with the chapter's own variance finding rather than ignoring the
tension between a "typical" week's actual cost and the smooth-sounding
long-run average.