It's Just Time

A coin pays +5% a flip. Where do you land?

Here is an offer any expected-value calculation tells you to take. You have €1,000, and you bet the whole pile on a fair coin — one hundred times in a row. Heads, your money grows by 50%. Tails, it shrinks by 40%. Average those two and every single flip is worth +5%: a positive expected value, a hundred times over. The textbook screams yes.

So before you read another word, commit. Drag the marker to where you think your €1,000 ends up after 100 flips — then lock it in. Don't peek at the answer first; the whole point is to catch your own intuition in the act.

Nobody lives the mean

If the crowd's average is rising while the typical player sinks, someone must be holding all that wealth. They are: a tiny sliver of players who hit an improbable streak of heads and ran away with everything. The mean is real — it just lives in a place almost no one reaches. Let time run longer and the gap doesn't close; it widens, until the average is the biography of a single lottery winner and the median is the biography of everyone else.

Below is the full crowd after a chosen number of rounds — every player's final wealth as a histogram (again on a log scale). Slide the number of rounds and watch the bulk of people slide left toward ruin while the mean marker flies right, off into a tail almost no one occupies. The readout tracks the share of all the wealth held by the luckiest 1%.

This is why the “expected return” of a multiplicative gamble can be a dangerous thing to plan a life around. It is an honest average over people, and you are not a crowd of people — you are one person, moving through time, and your timeline cannot borrow the winners' luck. The number that describes the room is not the number you will live. To act well you need the median's growth rate, the one that compounds along your single thread, not the mean's.

“The expectation value of wealth does not reflect what happens over time… It is dominated by rare events that almost no individual experiences.”

Ole Peters · The ergodicity problem in economics

If betting the whole pile makes the mean and the median fly apart, the cure is hiding in the bet size itself. Game Three turns that into the only dial that matters.

Don't bet the whole thing

The ruin in Game One was not caused by bad odds. The coin had a positive expected value the whole time. It was caused by exposure — betting all of your wealth, so that one bad run multiplies you down to nothing you can recover from. Keep the same coin, but stake only a fraction of your wealth each round, and the geometry changes completely: now there is a best fraction, and it is nowhere near all of it.

Below, the same +50% / −40% coin from Game One. The straight line is what expected value says: bet more, always, all the way to everything. The curved line is the growth rate you actually live through time at each bet size. Slide the fraction you stake and watch the two diverge — they recommend opposite things.

The peak is the Kelly fraction — the stake that maximises the growth you actually live. For this coin it sits at a quarter of your wealth, turning the whole-stake −5% a round into a steady positive climb. Notice what changed: not the coin, not the odds, only the exposure. Survival is a precondition for return, not a tax on it. This is the same lesson the barbell and via negativa tell from other angles — cap the downside first, and the upside takes care of itself over time.

“To make money, you must first survive… The ones who win in the long run are the ones who are still in the game.”

Nassim Nicholas Taleb · Skin in the Game