This is the graded finale for the whole course. It pulls the entire arc together: why wealth compounds multiplicatively and the expected logarithm of wealth is therefore the objective that matters, the ergodicity gap between the ensemble mean (a fantasy) and the time-average or median path (what you actually live), the derivation and use of the Kelly fraction , the growth hump that peaks at and collapses to zero near , why a player with a genuine edge can still bet themselves broke, why the honest move is fractional Kelly, and — just as important — where the model quietly lies. Several questions carry deliberate traps drawn from the most common misreadings of “maximise expected value,” so read each stem carefully before you lock in an answer.
How this exam works
This is a final, one-way exam. Questions come one at a time, and submitting an answer locks it for good — there is no going back, no retry, and no restart. Your score stays hidden until the very end, when you will see whether you passed. The pass mark is 70%. Some questions are marked select all that apply and need every correct option checked (and no wrong ones) to earn the point. Take your time on each question, because you only get one shot at it.
Why does personal wealth over repeated bets compound MULTIPLICATIVELY rather than adding up?
Select an answer to continue.
Course Recap
Big picture
The Kelly Criterion — the whole course
- The Kelly Criterion
- Multiplicative wealth & the right objective
- Staking a fraction of your bankroll makes wealth compound multiplicatively (win ×(1+fb), lose ×(1−f)); a single ×0 is unrecoverable, so the arithmetic average is the wrong target.
- Because factors multiply their logs add, so the growth rate of one path is the average log-factor — maximise E[log wealth], the time-average growth rate you actually live.
- The ergodicity gap
- Maximising E[wealth] on the 60/40 coin says bet everything (E[factor]=1+f(p−q) rises with f) — and that path is ruined with probability 1.
- On a multiplicative process the ensemble mean is inflated by a few lucky worlds while the median/typical path decays; Kelly maximises the time-average = median long-run wealth.
- Deriving & applying f* = p − q/b
- Bet your edge over the odds: f* = (pb−q)/b = p − q/b. Even money → f* = 2p−1 = your edge.
- Worked: 60/40 coin (b=1) → f* = 20%, growth ≈ +2.0%/round. p=0.5, b=2 → f* = 25%.
- The growth hump & over-betting
- g(f) = p·ln(1+fb)+q·ln(1−f): zero at f=0, peak at f*, back to ~0 near 2f*, negative beyond (60/40: f=40% → growth ≈ 0; f=1 → certain ruin).
- A positive-edge player can bet themselves broke; growth cost is roughly symmetric around f*, but consequences are not — erring LOW is far cheaper than erring HIGH.
- Fractional Kelly, ruin & honest limits
- Half-Kelly keeps ~75% of the growth for ~half the volatility; you overestimate your edge and full Kelly's drawdowns are brutal (~1/n chance of ever hitting 1/n), so bet a fraction — survival first.
- Limits: assumes known p and b; indifferent to short-run volatility and finite horizons; needs repeated, independent, divisible bets; NOT a licence to bet big.
- Multiplicative wealth & the right objective
Key takeaways — the whole course
The Kelly criterion answers one question — given a real edge, how much do you stake? — and its answer reframes the whole game. Personal wealth compounds multiplicatively, so the objective that matters is not expected wealth (an ensemble fantasy dragged up by a few lucky worlds that says “bet everything” and ruins you with probability 1) but the expected LOGARITHM of wealth, the time-average growth rate your single trajectory actually experiences. Maximise it and out drops f* = p − q/b: bet your edge divided by the odds — 20% on the 60/40 even-money coin, 25% on a 2-to-1 bet you win half the time. Long-run growth is a hump, not a ramp: it peaks at f*, collapses back to zero around 2f*, and turns negative beyond, which is why a player with a genuine edge can still bet themselves broke — over-betting buys less growth AND more ruin. Because the hump is flat on top but a cliff past the peak, erring low is cheap and erring high is catastrophic; because you never truly know your p and b and always overestimate your edge, and because full Kelly’s drawdowns are brutal, the honest default is FRACTIONAL Kelly (half or less), keeping ~75% of the growth for ~half the volatility. And respect where the model lies: it assumes known odds, it is indifferent to the volatility and finite horizons real people care about, it needs repeated independent divisible bets, and — read twice — it is not a licence to bet big. Its deepest teaching, from Kelly and Shannon in 1956 through Ed Thorp’s blackjack tables and hedge fund, is how easily a winning player goes broke. Survival first, growth second: bet your edge, and survive the variance.