Bet to Survive

I offer you a coin. It lands heads 60% of the time, and I pay you even money — bet a dollar, win a dollar on heads, lose your dollar on tails. You can play as many rounds as you like, staking any fraction of your money each time. This is a genuinely good bet: the edge is real and it is in your favour. So here is the only question that matters — how much of your pile do you put on each flip?

Your gut screams all of it. The coin is biased your way; more on the table means more profit; do the expected-value arithmetic and betting everything every round even maximises your average wealth. And yet if you bet everything, you are almost certainly going to end up broke — not unlucky-broke, but mathematically guaranteed broke, given enough flips. One tail, ever, and a bankroll that was multiplied down to zero can never climb back. The bet is favourable and the aggressive strategy is ruinous at the same time. That paradox is the entire subject of this course.

See it in one picture

Here is the paradox on a single chart. Three players face that exact 60/40 coin at even money. One stakes half the Kelly fraction (10% a round), one stakes the full Kelly fraction (20%), and one over-bets to double Kelly (40%). Each line is the typical (median) bankroll over 80 flips, drawn on a log axis so that steady compounding shows up as a straight climb. Press run a few times and watch the three fates separate.

Three things jump out, and each becomes a lesson. First, half-Kelly barely gives up any growth — it crawls upward almost as fast as full Kelly for far gentler swings. Second, full Kelly climbs fastest of the three, but only just, and its ride is stomach-churning. Third, the over-bet line spikes and then crashes: doubling the “optimal” bet doesn’t double your growth, it destroys it, because past a certain fraction a winning game turns into a losing one. There is a hump: growth rises with your bet size, peaks at Kelly, then falls back through zero at roughly twice Kelly and goes negative beyond. Bet past the far edge of that hump and a positive-edge game still grinds you to nothing.

Now flip the simulator’s readout from Typical player to Ensemble average on the over-bet setting. The average wealth suddenly looks spectacular — it can read in the millions — even though the median player is flat broke. That gap between what the average does and what a typical single player actually experiences is the reason “maximise expected wealth” is such dangerous advice, and it is the heart of Lesson 1.

What you’ll walk away with

By the end you’ll be able to look at any repeated, favourable bet — a trading position, a poker bankroll, a portfolio of risky projects — and answer how much to stake with a real number and a real reason, rather than a gut feeling that always errs toward “more.” Here is the ladder:

1. The coin flip that ruins you — how a positive-edge bet can wipe out almost everyone, why the ensemble average and the time-average growth of your wealth disagree, and why compounding means a single wipeout is permanent. This is the ergodicity punchline.
2. Maximising log wealth — the fix. Why maximising the expected logarithm of your wealth is what a single trajectory actually experiences, and how it hands you the Kelly fraction $f^* = p - \dfrac{q}{b}$, worked through on real numbers.
3. The growth curve — the hump in full: growth peaks at Kelly, returns to zero near $2f^*$, and turns negative beyond. Why over-betting and under-betting by the same amount cost the same growth, but only over-betting can kill you.
4. Fractional Kelly — why practitioners deliberately bet half Kelly or less: it keeps roughly three-quarters of the growth for about half the volatility, and it protects you from the fact that you always overestimate your own edge.
5. Transfer, and where the model lies — position sizing, bankroll management, venture and career bets, the Thorp/Shannon story, and the honest limits: Kelly assumes you know your edge, ignores short-run drawdown, and is not, ever, permission to bet big.

How to use this course

Every lesson opens with a quick guess, teaches the idea through a concrete bet and worked numbers, and checks that it stuck. Don’t skip the guesses — committing to an answer before you know is one of the most reliable ways to actually remember it. You’ll drive the bankroll simulator and the growth-curve chart from several angles and meet sorting and matching exercises along the way.

This is an expert-tier course, so it leans on a few models you’ve ideally met already. You’ll get the most from it if you’re comfortable with ergodicity and the time average (the difference between what an average of many players does and what one player experiences over time — the beating heart of Lesson 1), thinking in probabilities (reasoning about edges, odds, and long-run frequencies), and compounding (why multiplying your wealth round after round makes a single trip to zero unrecoverable). When you’ve finished the five teaching lessons, a graded final exam pulls it all together — it’s one-way, so once you submit an answer it’s locked.

Ready? The next lesson goes straight to the paradox — a coin that is genuinely in your favour, a strategy that maximises your average wealth, and why that strategy ruins almost everyone who follows it.