Probability and Expected Value: Lotteries, Insurance and Everyday Decisions
Expected value does not answer "what happens this time" but "what you average per attempt if you repeat it many times". It turns an uncertain choice into a comparable number — the most practical tool in probability.
The formula and a lottery example
EV = Σ (probability × net outcome).
A 10 ticket with a 1% chance of 500: winning nets 490, losing nets −10.
EV = 0.01 × 490 + 0.99 × (−10) = 4.9 − 9.9 = −5.
Repeat it long enough and you lose 5 per ticket. A single 500 win is possible, but it does not change the long-run average.
Where the house edge comes from
Bet 1 on a single number of a fair die:
- Win nets 5:
(1/6)×5 + (5/6)×(−1) = 0→ a fair game - Win nets only 4:
(1/6)×4 + (5/6)×(−1) ≈ −0.167→ lose about 0.167 per bet
A one-unit difference in payout turns break-even into steady loss. Every casino game trims the payout slightly and relies on volume to turn a small edge into certain revenue.
Independence and the gambler’s fallacy
After five reds in a row, the sixth is still the same probability — the wheel has no memory. Expecting black "because it is due" is the gambler’s fallacy.
Two useful calculations:
- Two sixes in two rolls:
(1/6)² = 1/36 ≈ 2.78% - At least one six in two rolls:
1 − (5/6)² ≈ 30.6%
For "at least one", use the complement. Independent events multiply; only mutually exclusive events add.
Why negative-EV insurance can still be right
Premiums must exceed expected claims, so insurance has negative EV. But people do not live on the average of money: one large loss can be unrecoverable, while a small certain premium is not.
That is risk aversion — utility is not linear. Expected value suits repeatable decisions with bearable losses; for rare, high-impact risks, ask whether you can survive the worst case instead.
Three common mistakes
- Judging a decision by one outcome: judge it by the information and EV available at the time.
- Ignoring low-probability, high-impact losses: EV can be dominated by extreme terms; looking only at the most likely outcome misses tail risk.
- Adding probabilities carelessly: only mutually exclusive events add directly; joint occurrence multiplies or needs inclusion–exclusion.
This article provides general information and calculation methods only. It is not medical, investment or legal advice; consult a qualified professional for your situation.