Give 61 people $25 and a coin rigged to land heads 60 percent of the time, and let them bet freely for 30 minutes. In the study at the center of this episode, 28 percent went bust, the average payout was $91 against a cap of $250, 18 players staked everything on a single toss, and two thirds bet on tails at some point. The fix has existed since 1956, when John Larry Kelly Jr. of Bell Labs worked out how much of a bankroll to wager on a favorable bet. For this coin the answer is exactly 20 percent of whatever you currently hold.

The episode explains why that fraction works: it maximizes the logarithm of wealth, which means the long-run compounding rate, and it can never reach zero. It then turns to the practical objections. Full Kelly betting produces brutal drawdowns, it assumes you know your true edge, and economists have argued for decades over whether maximum growth is the right goal for a person with a finite life.

• Without the $250 cap, steady 20 percent betting in the coin game would have produced an expected geometric mean wealth of more than $10,000.
• A 50 percent gain followed by a 50 percent loss turns $100 into $75, which is why sequential bets punish volatility and reward the geometric view.
• Many gamblers and investors bet half or a quarter of the Kelly amount as a margin of safety, because overestimating an edge leads to overbetting and ruin.
• Edward O. Thorp applied the formula to historical S&P 500 returns and got a fraction of 117 percent, implying borrowed money, a result that ignores margin calls and black swan crashes.
• Paul Samuelson objected on grounds of expected utility, an argument that reaches back to Daniel Bernoulli in 1738: a ruinous loss hurts far more than an equal gain helps.