Worked Examples
Four questions where the intuition and the arithmetic disagree — and the arithmetic is a single multiplication away.
Figures current · August 2026
E(X) = Σ (outcome × probability)per $2 ticket
Powerball jackpot odds are 1 in 292,201,338, and a ticket costs $2. Multiply a $100 million jackpot by that probability and you get what the jackpot slice of the ticket is actually worth.
$0.34of expected jackpot value at a $100M jackpotticket: $2.00
→Even ignoring taxes, shared jackpots and the smaller prize tiers, the jackpot alone would have to pass $584 million before a $2 ticket broke even.
P(A and B and C) = P(A) × P(B) × P(C)if independent
The S&P 500 finished lower in 26 of the 97 calendar years from 1928 through 2024 — a base rate of about 27%. Multiply that by itself three times and you have three straight losing years.
1.9%three down years in a rowone down year: 27%
→The multiplication rule only holds if the years are independent, and market years are not — which makes this an estimate, never a forecast.
P(at least one) = 1 − P(none)over 10 years
Same 27% base rate, opposite question. The chance of a decade with no down year at all is 0.73 raised to the tenth power, so subtract that from one and you have the chance of at least one.
95.6%chance of a down year within tenno down years: 4.4%
→One base rate makes a three-year streak rare and a clean decade nearly impossible — all that changed was the question.
68% within ±1σ, 95% within ±2σthe empirical rule
Take a fund whose yearly returns average 8% with a standard deviation of 15 points. If those returns are normally distributed, about 68 years in every 100 land somewhere between −7% and +23%.
68%of years inside ±1σ95% inside ±2σ
→Standard deviation turns "risky" into a number — the wider the curve, the more ordinary a terrible year becomes.
Sources: Powerball published jackpot odds; NYU Stern (Damodaran) annual returns on the S&P 500, 1928-2024. Reviewed August 2026.
