Probability and statistics
Expected value
Weigh each outcome by its probability and add them up; that is what a repeated bet is worth on average, and only on average.
Christiaan Huygens, whose De ratiociniis in ludo aleae was the first printed work on probability, written after he learnt of the Pascal and Fermat correspondence in Paris in 1655; Jacob Bernoulli extended it in Ars Conjectandi (1713).
The expected value of a gamble is the sum of each payoff times its probability. A ticket that pays $100 one time in a thousand is worth 10 cents; if it costs $2, every ticket loses $1.90 on average, and a million tickets lose close to $1.9 million for certain. The number tells you what the bet is worth if you could take it many times, which is why casinos, insurers and lotteries are built on it: every bet on a double-zero roulette wheel carries an expected loss of 5.26 percent, and the house collects that over millions of spins. Huygens worked the idea out for games of chance after learning of Pascal and Fermat's letters, and Bernoulli's Ars Conjectandi used it to say how much a player should expect to win at various games.
The same arithmetic finds bets worth taking. Massachusetts ran a lottery called Cash WinFall from 2004 in which, whenever the jackpot reached $2 million with no winner, it 'rolled down' into the smaller prizes, making them five to ten times larger than usual. In a roll-down draw a $2 ticket was worth more than $2 in expectation, and three betting groups, one of them MIT students, spent about $40 million on tickets over seven years and won about $48 million. The MIT group found it had to buy at least $600,000 of tickets in a single roll-down for its average profit of more than 15 percent to be dependable: at that scale the law of large numbers turned an expected gain into an actual one.
Expected value ignores two things. It ignores variance: two bets with the same mean can behave very differently, and a positive-expectation bet you can take only once is still a gamble. And it ignores ruin. A bet that returns 50 percent on average but wipes you out one time in ten has a positive expected value and a certain end if you keep taking it, because you cannot take the next bet from zero. That is why bet size, margin of safety and the shape of the downside matter as much as the mean, and why the syndicates bought in bulk rather than a ticket at a time. Use expected value to rank options you can repeat; for one-shot bets that can end you, look at the worst case first.
Example
In July 2012 the Massachusetts inspector general, Gregory Sullivan, reported on Cash WinFall, a $2 lottery game introduced in 2004. When the jackpot reached $2 million without a winner it rolled down into the lower prize tiers, and in those draws the tickets had a positive expected value. Three groups, one of them MIT students trading as Random Strategies, spent about $40 million on tickets over seven years and won an estimated $48 million; the MIT group alone put in roughly $17 million to $18 million and took out more than $3.5 million in profit. The report found that lottery officials had known for years and had bent rules on bulk sales, and the game was discontinued in 2012.
Where it applies
Price an A/B test, a retry policy or a cache by expected cost per request, but check the tail: a 0.1 percent chance per day of a $100,000 outage is $100 a day in expectation and still something you insure against. When an AI agent takes paid actions, expected cost per task is what you budget on and the worst single run is what you cap.
Think about it
Which decision this week are you making on the average outcome when it is really a one-shot bet whose worst case you have not priced?
Applied it to something real? Log it.
Test yourselfA $2 lottery ticket pays $100 with probability 1 in 1,000 and nothing otherwise. What is it worth, and when would buying many be rational?
Its expected value is 10 cents, so each ticket loses $1.90 on average. Buying is rational only when the expected payout exceeds the price, as in Cash WinFall roll-downs, and only in enough volume for the average to show up.
Related models
- Probabilistic thinkingThink in odds rather than certainties: estimate how likely each outcome is, and judge a decision by those odds rather than by the single result that happened.
- Asymmetric betsTake bets where the most you can lose is small and known and the most you can win is large, even if you expect to lose most of them.
- Margin of safetyLeave a wide gap between what you expect to need and what you build or pay, so that your errors and bad luck do not sink you.
- The law of large numbersAverages of many independent trials settle towards the true value; averages of a few do not, and the law never 'corrects' a streak.
- Fat tailsIn some domains the rare extreme event is so large that it dominates everything else, and the bell curve cannot see it coming.