A decision can be sound and still lose. That is not a paradox. Poker involves hidden cards and chance, so a good choice only changes the odds in your favor; it does not decide the outcome. Variance is the name for how far results typically spread around the long-run average.
Good decisions and bad results
Suppose a player makes a decision that wins 80% of the time. The remaining 20% is a loss, and nothing about the decision was wrong. After five such decisions, the chance of at least one loss is 1 − 0.8^5 = 1 − 0.32768 ≈ 67%. Losing at least once in five trials is the usual outcome, not a surprise.
Now take a smaller edge. A bet wins 55% of the time and loses 45% (assume even stakes, ignoring fees, purely as an illustration). Over many repetitions it is favorable. Yet the chance of losing five in a row is 0.45^5 ≈ 0.018, about 1.8%. That is rare on any given run of five, but across thousands of hands, runs like it are likely to turn up somewhere.
A simple simulation
The easiest way to see variance is to repeat a coin-style experiment. Imagine 100 independent situations, each won with probability 0.55. If you repeated those 100 situations many times, say in a spreadsheet, the number of wins in each run would vary.
- The average is 100 × 0.55 = 55 wins.
- The standard deviation is the square root of 100 × 0.55 × 0.45 = 24.75, which is about 5.
- Roughly two thirds of runs would land within one standard deviation of the average, so between about 50 and 60 wins.
- Roughly 95% would land within two standard deviations, between about 45 and 65.
These ranges use a normal approximation, so treat them as estimates. They imply that a run with 46 wins, though below the 50-win mark, would not be remarkable. The probability of ending with fewer than 50 wins (losing more than half) is about 13%, roughly one run in eight, even though every single situation was favorable. Here the player is favored every time and still finishes behind in about one run in eight.
Why a longer sample helps
Now increase the number of situations from 100 to 400. The average is 400 × 0.55 = 220 wins. The standard deviation is the square root of 400 × 0.55 × 0.45 = 99, about 10.
In raw counts, the spread has doubled, from 5 to 10. But relative to the number of trials it has halved: 5 out of 100 is 5%, while 10 out of 400 is 2.5%. Quadrupling the sample halves the relative spread. This is the content of the law of large numbers: the share of wins tends to settle near 55% as trials grow, while the raw gap from the expected count can still grow. It also explains why the number of observations, the sample size, matters so much when judging results.
The practical consequence is slow feedback. If an edge is small, many hundreds or thousands of hands may be needed before results begin to reflect it, and even then they are only an estimate. Judging a decision by one session, or one hand, mixes skill with luck in unknown proportions.
Variance depends on the situation
Variance is not the same for every style of play or every type of decision. A decision where you win a small amount often and lose a large amount rarely behaves differently from one with the reverse pattern, even if their averages match. Higher-variance situations produce wider swings, and the average takes longer to show. Discussions of volatility and return make this distinction: two choices can have the same average but quite different spreads.
What variance means for money
Large swings are normal, and bankrolls have to survive them. A run of 10 or 20 losses, which may be entirely within normal variation for a modest edge, can wipe out a small bankroll before the average has had time to assert itself. This is the idea behind risk of ruin: the probability of running out of money before the long run arrives. It is also why chasing losses is harmful. After a bad stretch, the odds on the next hand have not shifted in your favor, and raising the stakes to recover only increases exposure.
Note too that everything above assumes a favorable edge. If the game has negative expectation, as casino games with a house edge do, more trials make the average loss more certain, not less. And an edge in poker is not guaranteed: it depends on opponents, on rake, and on how accurately the player judges their own skill.
Reading results fairly
A few habits help keep results in proportion:
- Judge a decision by the information available when it was made, not by the cards that followed.
- Avoid inferring skill or the lack of it from a short sample.
- Recognize that streaks, good and bad, are expected from randomness alone.
- Do not borrow or raise stakes in response to losses.
None of this makes gambling safe. Outcomes are uncertain, fees reduce average results, and money can be lost. Anyone whose play is causing distress should consider contacting a local support service.