never confuse the average with your own path

There is a deep difference between what happens on average across a thousand people and what happens to one person living through a thousand rounds, and they only match under conditions real life rarely meets. Flip a coin. Heads, your money grows 50 percent. Tails, it falls 40 percent.

\[ \begin{aligned} \text{average of the two:}\quad & \tfrac{1}{2}(1.5) + \tfrac{1}{2}(0.6) = 1.05 \\ \text{multiplied in a row:}\quad & 1.5 \times 0.6 = 0.90 \end{aligned} \]

the average of a 50 percent gain and a 40 percent loss is a 5 percent gain (1.05). but you don't collect the average, you collect the two multiplied one after the other, and 1.5 × 0.6 = 0.90 leaves you down 10 percent. gains and losses multiply, they don't average.

after 100 rounds, about 86% of players end up poorer than they started; the 132x average is created by the lucky few out in the right tail.

each bar is how likely one player is to finish at that wealth after 100 rounds. gray means ending poorer than you started, blue means coming out ahead. the average sits far out in the tail, describing almost no one.

A bet that looks excellent averaged over a crowd can still ruin the person who keeps taking it, because that person gets one trajectory -- and a single ruin ends the game. This is why avoiding catastrophe matters more than chasing the highest expected return. You have to survive every round to ever see the average of them.