20. The mathematics of luck

The tip as a probabilistic experiment

Every bet is a weighted coin flip: it wins with probability p, loses with probability 1 − p, and nothing more happens. A tip series is therefore a binomial sequence of experiments, whose behaviour is not a matter of opinion but of formulas: from n tips with a hit chance of p, the expected number of hits is n × p, and its standard deviation is sqrt(n × p × (1 − p)).

In numbers: from 100 tips at 50% you expect 50 to hit, with a deviation of sqrt(100 × 0.25) = 5. Your number of hits will, in the overwhelming majority of cases, be in a band of ±2 deviations around the expected value: anything between 40 and 60 is completely normal, without any change in skill. What bettors experience as "getting into form" or "shattered confidence" is largely this band — maths, not story.

Variance is not a flaw in the system but the system itself. Whoever does not know their own deviation manufactures an explanation for every swing, and every explanation costs money.

The deviation of the tip series in money

Even more important than the number of hits is how much your money fluctuates. With a unit stake a bet’s profit is either odds − 1 or −1, and its deviation can be calculated too. The simplest case: 2.00 odds, 50% chance, zero EV. A single bet’s deviation is 1 unit, that of 100 bets sqrt(100) = 10 units.

This means that after 100 zero-EV bets your result can be anywhere between ±20 units (±2 deviations): you can "measure" a +20% ROI while having no edge whatsoever, and a −20% while having done nothing wrong. So a 100-tip sample is not proof but noise, and you now know this not from feeling but from a formula.

Remember the magnitude: 100 tips at odds around 2.00 give a ±20% ROI swing for free, without skill. Whoever draws a conclusion from less than this is reading the noise (chapter 27).

Odds level and variance: 1.50 vs 5.00 with the same edge

Two players, both betting with +5% EV, at a unit stake. One at odds of 1.50 (70% chance), the other at 5.00 (21% chance). In the long run they head to the same place, but their journeys differ dramatically:

The same edge comes with triple the deviation at high odds: reaching the same certainty needs roughly nine times as many tips. This is why the longshot player feels forever on a rollercoaster, and why many a winning strategy is abandoned by its owner before it can prove anything (chapter 21).