Week 4: Expected value and variance

Formalizing EV beyond one hand, why a correct call still loses most of the time it's made, and why sample size is doing more work than it looks

15 March 2027

Formal EV#

Week 3 used EV(call) = equity × pot − (1 − equity) × call for one specific hand without naming the general pattern behind it. In general, for a decision with outcomes worth x₁, x₂, … xₙ occurring with probabilities p₁, p₂, … pₙ:

EV = Σᵢ pᵢ · xᵢ

Week 3’s formula is this sum with exactly two outcomes: win the pot, or lose the call. Everything below stays inside that same two-outcome shape, because it’s the shape a single street’s calling decision always has.

A profitable version of the flagship hand#

Reuse week 3’s flush draw (9 outs, 19.6% equity, one card to come), but raise the stakes so the call is actually worth making: pot $300, call $50.

Required equity = 50 / (300 + 50) ≈ 14.3%. Hero’s 19.6% clears that comfortably:

EV(call) ≈ +$18.48 — a genuinely profitable call, this time.

Why a correct call still loses most of the time#

“Profitable” describes an average over many repetitions, not this one hand. This call wins $300 only 19.6% of the time; the other 80.4% of the time it loses the $50 call outright. The spread between those two outcomes — not just their average — is what variance measures:

This call’s standard deviation is about $139, roughly 7.5 times its own $18.48 edge.

A correct, positive-EV call losing money on any single attempt isn’t bad luck defeating good strategy — the numbers above say it’s the expected short-run experience of a play whose edge is small relative to its spread.

The sample-size fallacy#

If variance is large relative to EV, then judging a play from a handful of results is judging noise. Concretely: this hand’s true win rate is 19.6%. The standard error of an observed win rate over N repetitions is

√(p(1−p) / N)

At N = 10 repetitions, that’s about ±12.5% — wide enough that a perfectly correct play can look badly wrong, or a bad one can look great, purely from the sample size. At N = 300,000 — the trial count week 1’s harness actually used for its equity claim — the same formula gives about ±0.07%, which is why that number was trustworthy and ten hands’ worth of results never would be.

Risk-adjusted EV#

Raw EV treats a $1 win and a $1 loss as exact opposites, which is a fine assumption for a single hand and a bad one for a whole session or a whole bankroll: losing a stack you can’t rebuy hurts more than the mirror-image win helps. One common formalization discounts EV by how much variance comes with it — a certainty equivalent of EV minus some multiple of variance, for a risk-aversion parameter chosen by the player’s actual stakes and bankroll. Week 11 makes a specific, principled version of this precise, once there’s a bankroll to protect and a growth rate to optimize instead of a free-floating risk parameter.

Before next week#

The variance figure above was for one hand in isolation. Week 5 changes what’s being measured — not the spread of one decision’s outcomes, but the value of the information a player in position gets to see before making it.