www.astralcodexten.com/p/kelly-bets-on-civilization
1 correction found
If you just bet $1 each time , then (on average), you’ll only have $1,008.
This expected-value calculation is off. Under the article’s own assumptions, five $1 bets have an expected total gain of $2.50, so the expected bankroll is $1,002.50, not $1,008.
Full reasoning
Using the game setup stated in the post, a $1 bet on a double-or-nothing coin flip with a 75% win chance has two outcomes: +$1 with probability 0.75 and -$1 with probability 0.25. Expected value is computed by multiplying each outcome by its probability and summing them.
So the expected gain from one $1 bet is:
- $1 × 0.75 = +$0.75
- -$1 × 0.25 = -$0.25
- Total expected gain per bet = +$0.50
Over five such bets, linearity of expectation gives an expected total gain of 5 × $0.50 = $2.50. Starting from $1000, the expected bankroll after five $1 bets is therefore $1002.50.
So the article’s figure of $1,008 is a math error under the assumptions it gives. (By contrast, the preceding “bet everything” figure of about $7,500 is a reasonable rounding of the correct expected value, $7,593.75.)
2 sources
- 7.11 Expected Value - Contemporary Mathematics | OpenStax
Expected value: If O represents an outcome of an experiment and n(O) represents the value of that outcome, then the expected value of the experiment is: ∑ n(O) × P(O).
- A Course in Networks and Markets: Game-theoretic Models and Reasoning
One nice property of expectation is that the expectation of the sum of random variables is the sum of the expectations.