The Bicycle Guide to The Mathematics of Poker

1. 1Define the bankroll as a fixed, designated amount, and define ruin as losing all of it. The source models risk of ruin as the chance of losing a fixed amount in a game with positive expectation but some variance, so the number is meaningless until the amount is fixed in writing.
2. 2Pull your own hand histories. Online sites let you request complete histories, usually including mucked hands and all actions, and commercial database programs manage them. Use this record, not memory, as the source for your results at each game level.
3. 3Estimate two inputs per game level from that record: your win rate and your variance (the spread of your session or hand outcomes). Risk of ruin is driven by the relationship between these two and the bankroll. Higher variance against the same win rate and bankroll means more ruin risk.
4. 4Treat the win rate as an estimate with its own uncertainty. The source extends the risk of ruin model to include the uncertainty surrounding any observation of win rate. Even a seemingly large sample, such as 10,000 hands, is often too small to pin down a win rate, so do not plug in a point estimate as if it were known.
5. 5Check the sample for selection bias before trusting it. The source warns that collected data often carries significant selection bias, and that gathering raw data is the easy part. Ask whether the hands you are viewing are a representative slice of your play, or only one game, stake, or stretch of running good.
6. 6Choose the game level and bankroll together. The source lists choosing an appropriate game level as a bankroll-theory topic alongside risk of ruin, the Kelly criterion, and certainty equivalent. Lower the stake, or add bankroll, until the ruin probability from your conservative (not best-case) inputs is acceptable to you.
7. 7Hold the premise the source uses for all play: you are adequately bankrolled for the game you are in. If you cannot honestly say that, fix the bankroll or the game level before working on strategy. Maximizing expected value only pays off if you survive to realize it.
8. 8Re-run the estimate as data accumulates and whenever you change games. Hand-history databases make this cheap. Update the win rate and its uncertainty rather than keeping an old number.

The source describes a player who gathers 10,000 hands on an opponent to identify that opponent's win rate, then warns that this relatively large number is often quite insufficient to assess the situation, and that the hands viewed may be biased. Turn the same lens on yourself. You export your own histories from a site and your database shows a positive win rate at your current level. Before moving up, you ask two questions the source raises. First, is 10,000 hands enough to treat this win rate as true? The source says a sample like this often is not, so you carry a range of plausible win rates, not one number. Second, are these hands unbiased, or all from one favorable stretch or game type? You then run the risk of ruin model with the low end of your win rate range and your observed variance, because the source's extended model exists to account for exactly this uncertainty. If the low-end inputs give an unacceptable chance of losing the whole designated bankroll, you stay at the current level or add bankroll. If the low end is still acceptable, moving up is defensible. The decision rests on the pessimistic estimate, not the raw database number.

• If your win rate comes from a small or unrepresentative sample, then use a conservative (lower) win rate in the risk of ruin calculation, since the source shows raw hand counts are often insufficient and biased.
• If the ruin probability at a level is unacceptable under conservative inputs, then drop the game level or add bankroll before playing it. Do not rely on strategy to compensate.
• If variance rises (a different game type or format) while win rate and bankroll stay the same, then treat ruin risk as higher and re-estimate before continuing.
• If you cannot say you are adequately bankrolled for the game, then stop treating decisions as pure EV maximization. The source's EV-first approach assumes adequate bankroll as a premise.
• If the money at stake is not meaningful to you in a linear way (the source notes preference for money is non-linear), then set your acceptable ruin probability lower than a pure-math view would suggest.

• Treating a database win rate as certain. Tell: you plug one number into the model and never state a range, though the source says even large samples often cannot settle a win rate.
• Mistaking easy data collection for finished analysis. Tell: you feel done once the database is full, without checking selection bias or sample adequacy.
• Leaving the bankroll undefined. Tell: you cannot name the exact amount whose loss counts as ruin, so risk of ruin has no fixed amount to refer to.
• Sizing the bankroll to win rate alone and ignoring variance. Tell: two games with different swings get the same bankroll requirement.
• Never updating after the first estimate. Tell: your stakes decision rests on an old sample even though your database keeps growing.

• ▸Designated bankroll (amount): write the exact sum whose total loss counts as ruin.
• ▸Game and level: name the specific game and stake being evaluated.
• ▸Hands in sample: count the hands in your own database for this level.
• ▸Observed win rate: record the measured figure from your hand histories.
• ▸Conservative win rate: enter a deliberately lower figure reflecting sample uncertainty, since the source says samples are often insufficient.
• ▸Observed variance: record the spread of your results at this level.
• ▸Selection-bias check: note whether the sample covers varied games and periods or only a narrow slice.
• ▸Computed risk of ruin: result of the risk of ruin model using the conservative win rate.
• ▸Decision: stay, move down, add bankroll, or move up, with the next date you will re-run this check.

1. 1Split every decision into two stages and keep them separate. Stage 1 gathers information: opponent tendencies, your cards, pot size, stack sizes. Stage 2 synthesizes that information into an action. Intuition and reads belong in stage 1 only. In stage 2 you calculate.
2. 2Write your assumptions down as explicit inputs before you compute: how the opponent plays, what he can hold, what you hold, the pot, the chips left. The source's point is that every player already runs an implicit model, because calling, raising or folding states a belief about the relative value of those actions. Writing the model down makes it checkable.
3. 3Replace 'check or bet' with plans that include your response to what comes next. The source's stud example lists five: check-fold, check-call, check-raise, bet-call, bet-fold. Many errors are prevented by deciding what you will do after the opponent acts before you take the first action.
4. 4Calculate expectation for each plan and take the highest. Where later action ends the betting, the form is <plan> = P(you win) × (new pot size) − (cost of the action on this street). Start with the branches that need no further modelling. Cases where the opponent's response gives the same value whatever your intention can be dropped from the comparison.
5. 5Correct your probability inputs deliberately. The source says the brain is poor at estimating probabilities, especially low-frequency ones. Compute them instead of eyeballing them, and treat any gut number for a rare event as suspect.
6. 6Think in strategies, not isolated decisions. Specify what you do with every hand at every decision point, then test whether you are playing balanced. If you mix, each way you play a given hand should have the same expectation against the opponent's optimal strategy. Do not sacrifice immediate expectation for deception, because optimal play has no 'loss leaders'.
7. 7Choose your mode explicitly. With a defensible assumption about the opponent, play exploitively and let the math act on that assumption. Without one, play optimally or pseudo-optimally, so no strategy can punish you. In either mode the action comes from calculation, not feel.
8. 8Review afterwards by asking which assumption was wrong, not whether the result was good. The calculation is only as good as the assumptions it consumed, so errors get fixed at that input.

The source poses Example 7.1: $30-$60 seven-card stud, pot $400. Player X shows A-K-Q-J-T. Player Y shows A-T-8-5-2 and has only $120 left, with X covering him. The passage sets up the analysis but does not give a final solution, so the structure is what you copy. First, X does not choose between two options. X chooses among five: check-fold, check-call, check-raise, bet-call, bet-fold. Second, X can evaluate two of them at once. If Y raises, there is no more betting on later streets, so bet-call is worth <X, bet-call> = P(X wins) × (new pot size) − (cost of action on this street). Bet-fold is the same calculation with the folding outcome. Third, the cases where Y folds or calls give X the same value whatever X intended against a raise, so they can be set aside when comparing bet-call and bet-fold. Fourth, X compares all five values and picks the highest. Fifth, the source warns that in such spots the correct play can be 'neither obvious nor intuitive'. That is the reason to compute instead of trusting a feeling about the strong hand.

• If you are about to act on a hunch, then first write down the assumption behind it and compute the expectation under that assumption. Act on the number.
• If you cannot defend an assumption about how the opponent plays, then default to optimal or pseudo-optimal play. Do not invent a read.
• If you plan to mix actions with a hand, then each way of playing it must have equal expectation against the opponent's optimal strategy. If one line is worse, shift the hands to the better one.
• If you are tempted to make a play that gives up expectation now for deception, then drop it. Optimal strategies do not contain 'loss leaders'.
• If the spot is a single street with a bet available, then evaluate check-fold, check-call, check-raise, bet-call and bet-fold, and choose the highest-expectation plan.

• Letting intuition choose the action. Tell: you can name a feeling but not an expectation figure for each option.
• Treating 'it depends' as the answer. Tell: your reasoning ends at 'depends on the opponent' with no assumptions listed and no calculation after them.
• Comparing only 'bet' versus 'check' and ignoring your follow-up. Tell: you cannot say what you will do when raised.
• Trusting gut probabilities for rare events. Tell: your equity estimates are round numbers you never calculated.
• Mistaking the book's stance for a rejection of psychology. Tell: you discard useful reads altogether. The source treats psychology as valuable for exploitive play and as the origin of the assumptions the model uses; it simply does not let intuition choose the action.

• ▸Situation: street, pot size, chips behind, who covers whom.
• ▸Stage 1 assumptions: how the opponent plays and what he can hold, written as explicit statements.
• ▸Your hand and the probability you win in each branch, calculated, not guessed.
• ▸Plan list: expectation for check-fold, check-call, check-raise, bet-call, bet-fold, using P(win) × new pot − cost of action on this street.
• ▸Branches skipped: cases where the value is the same whatever your intention, and why.
• ▸Chosen plan: the highest-expectation option.
• ▸Mixed-strategy check: if you mix, do all ways of playing this hand have equal expectation? Y/N.
• ▸Mode: exploitive (name the assumption) or optimal/pseudo-optimal (no usable read).
• ▸Post-hand audit: which assumption, if any, proved wrong?

1. 1Replace expectation with utility as the decision metric. The source says that for real-world decisions you cannot use expectation alone; you must assign a utility function and evaluate outcomes in terms of utility. For bankroll decisions, use U(x) = ln x, the Kelly criterion form, where x is your total bankroll after the outcome.
2. 2Write each candidate game as a set of outcomes with probabilities: the result of a session or unit of play, and how likely each result is. Each game needs an explicit expectation (win rate) and a measure of dispersion. The source tracks variance, standard deviation, z-scores, and win-rate-based variance as the dispersion statistics.
3. 3Substitute your current bankroll into the log utility for each game. For each game, compute the probability-weighted ln(bankroll + outcome) across its outcomes. The game with the higher expected utility at that bankroll is the correct one to play.
4. 4Find the crossover bankroll between a smaller and a larger game. The source states that when the games have appropriate conditions, there is a bankroll level above which you play the larger game and below which you play the smaller. Rerun step 3 at several bankroll levels until the preferred game flips. That level is your move-up and move-down line.
5. 5Score every option by its Sharpe ratio, expectation divided by standard deviation. The source says that when investments are infinitely scalable, maximizing the Sharpe ratio gives the best risk-adjusted return. Compute it for each game and each combination of games.
6. 6Reduce volatility only through real diversification. Two methods are named: swapping action with other players, in tournaments or ring games, which cuts volatility while preserving Sharpe, even when your swap partner has the higher expectation; and multi-table online play, for example two $10-$20 games instead of one $20-$40 game, which gives similar expectation with lower variance. Use the second only if you can concentrate adequately on several tables.
7. 7Do not count playing different game types (sit-and-gos, multi-table tournaments, limit ring games) as variance reduction. The source's reasoning is that owning two stocks means owning both 100% of the time at half size each, whereas switching games is like holding one stock half the time and then swapping to another. Keep variety for its other benefits: exploiting temporary situations, such as a very weak, very rich player at his favorite game, and building a broader view of poker.
8. 8Treat the bankroll model as a function of your assumptions. Once you have turned your reads into assumptions about win rate and variance, the source says the mathematics takes over and intuition stops being a reliable guide. Update the inputs as your data changes, and let the model choose the action.

The source's concrete variance-reduction case is two $10-$20 games versus one $20-$40 game. Say the single $20-$40 game gives you win rate W and standard deviation S, so its Sharpe ratio is W/S. Split the same attention across two $10-$20 games. The source states the result: similar expectation with lower variance, so the Sharpe ratio rises. The caveat in the text is that you might have been able to run two $20-$40 games instead, so the smaller stakes can sacrifice equity. Resolve that trade-off with the log-utility test. Compute expected ln(bankroll + result) at your current bankroll for (a) one $20-$40 game, (b) two $10-$20 games, and (c) two $20-$40 games if you can concentrate adequately on that many tables. Play the highest. As your bankroll grows, the preferred option moves toward the larger game, and below the crossover bankroll it moves back down. Use your own measured W and S; the source supplies the method, not the numbers.

• If your bankroll is above the crossover level where the log-utility of the larger game exceeds the smaller, play the larger game. If it is below, play the smaller.
• If two options have similar expectation, choose the one with the higher Sharpe ratio (lower standard deviation).
• If you can concentrate well on multiple tables, then split your action into more tables at lower stakes to cut variance. If concentration degrades, don't.
• If a swap partner offers action exchange, then take it when it lowers your volatility and keeps your Sharpe ratio, even if the partner has the higher expectation.
• If your reason for adding a new game type is portfolio risk reduction, then don't. Add it only to exploit a temporary edge (such as a weak, rich player in his favorite game) or to broaden your understanding.

• Judging games by expectation alone. The tell: you move up in stakes whenever the win rate is higher, with no bankroll threshold or variance figure behind the move.
• Treating game variety as diversification. The tell: you rotate among sit-and-gos, multi-table tournaments, and limit ring games and expect your bankroll swings to shrink. They don't, because you hold each 'stock' only part of the time.
• Using intuition to pick the action after the assumptions are set. The tell: the calculation says one thing and you play another because it feels right. The source says to use intuition for gathering information, not for the choice.
• Ignoring how poorly people estimate low-frequency probabilities. The tell: you underestimate how often bad runs occur, so your bankroll feels fine until a routine downswing exposes it.
• Wrongly assuming a higher-expectation player gains nothing from swapping. The tell: you refuse action swaps with weaker players, even though the source shows swapping can improve your Sharpe ratio.

• ▸Current bankroll (B): your total bankroll today, in the same units as the outcomes.
• ▸Candidate game A (smaller): stakes and format.
• ▸Candidate game B (larger): stakes and format.
• ▸Win rate (expectation) for each game: your estimated result per unit of play, from your own records.
• ▸Standard deviation per unit for each game: dispersion of your results, measured from your records.
• ▸Sharpe ratio (expectation / standard deviation) for each game: compute and compare.
• ▸Expected ln(B + outcome) for each game: probability-weighted log utility at your current bankroll.
• ▸Crossover bankroll: the B at which the two games' expected utilities are equal; play A below it and B above it.
• ▸Volatility-reduction options: note swap partners available and whether multi-tabling at lower stakes is feasible (concentration check: yes/no).

1. 1Split every decision into two stages, as the source does. Stage one gathers information: opponent tendencies, your cards, pot size. Stage two picks the action. Stage two is arithmetic, so intuition has no role in it.
2. 2Write down your assumptions before computing anything: the opponent's distribution (which hands he bets with, and how often he bluffs), the pot, and the bet size. The EV you get is only as good as these inputs, so each one needs to be explicit.
3. 3Compute the expectation of every available action (call, fold, raise) as probability-weighted outcomes. Use the source's form: EV(call) = p(opponent value hand) × (lose one bet) + p(opponent bluffing) × (pot + one bet). Then pick the maximum.
4. 4Solve for the break-even frequency. Set the EV to zero and solve for the unknown, usually the opponent's bluffing or calling frequency. This gives a threshold to compare against your read, not a verdict from gut feel.
5. 5Treat reads as probability updates, not overrides. The source says playing mathematically incorporates reads more accurately than playing by feel, and Bayes' theorem quantifies how far new information should move your estimate. Never abandon the math because you have a 'read'; put the read into the inputs.
6. 6Discount your confidence in the inputs. A 1,000-hand sample yields perhaps 100 showdowns, spread across positions and flop textures. Rate each estimate by how little direct evidence supports it, and prefer the action that holds up if the estimate is off.
7. 7Evaluate the whole strategy, not just the single decision. Preflop, flop, turn and river play are parts of one strategy. When you can, test a candidate line by asking how a maximally exploiting opponent would punish it.

The source's river example: the pot is $655, Player B can bet $80, and A is deciding whether to call. B has a flush 20% of the time, so calling loses $80 in that case. B bluffs with frequency y, and calling wins the pot plus the bet, $655 + $80 = $735. So <A, call> = (0.2)(-$80) + (y)($735) = $735y - $16. Setting this above zero gives y > about 2.2%. At exactly that bluff frequency, A's call has EV 0 and A is indifferent between calling and folding. If B bluffs more than 2.2%, A calls every time. The same method applies to B: <B, bluff> = x(-$80) + (1-x)($655) = $655 - $735x. This is positive when A calls less than about 89.1%, and at 89.1% B is indifferent to bluffing. The decision-quality point: the right call is determined by comparing your estimate of y with 2.2%. If you believe B bluffs even a few percent of the time, calling is correct. Each extra percentage point of bluffing adds $7.35 to the value of calling, which is why a skeptic who always calls profits from B's bluffs.

• If your estimate of the opponent's bluff frequency is above the break-even threshold (2.2% in the river example), then call, and do so every time. If it is below, fold.
• If an action's EV is zero at the opponent's frequency, then you are indifferent. Do not agonize; choose by whichever other consideration matters, such as how your read could be wrong.
• If you catch yourself preferring a play because of a 'feel' that contradicts the calculation, then convert the feel into a changed input (a new bluff or call frequency) and recompute. Do not override the output.
• If your read comes from a small sample (a few showdowns in 1,000 hands), then widen your range of plausible frequencies and choose the action that is positive across that range.
• If a mistake-prone opponent looks exploitable, then exploit him now, but expect it to be temporary. Opponents adapt and reduce their clear errors.

• Letting intuition choose the action: you justify a call or fold with 'he's bluffing' or 'numbers go out the window.' The tell is that you can't state the EV of the options or the frequency you assumed.
• Overstating certainty about opponent reads: you treat a player classification as fact after a modest sample. The tell is a precise-sounding frequency based on only a handful of shown-down hands.
• Judging decisions street by street in isolation: you ignore that preflop, flop, turn and river lines belong to one strategy. The tell is a decision that is fine alone but makes your overall range easy to exploit.
• Confusing results with quality: you assess a call by whether the bluff happened. The tell is that you change your play after one outcome without changing any input frequency.
• Assuming a permanent edge from past observations: you keep exploiting a tendency after the opponent has adapted. The tell is that your once-profitable exploitation stops working while your estimates remain unchanged.

• ▸Situation: street, pot size, and bet size you face or consider.
• ▸Opponent range assumption: the hands and share of his range that are value versus bluffs.
• ▸Source of read and sample size: where the estimate comes from and how many showdowns support it.
• ▸Action options: list call, fold, raise, and any other candidates.
• ▸EV formula per action: probability-weighted outcomes, written out (e.g., p(value)×loss + p(bluff)×(pot + bet)).
• ▸Break-even frequency: set the EV to zero and solve for the unknown frequency.
• ▸Your estimate vs. break-even: where your read falls relative to the threshold, with a plausible low-high range.
• ▸Chosen action: the highest-EV option, and whether it holds across the whole range.
• ▸Post-hand review: which input (not the outcome) proved wrong, and the update you make.

1. 1Define the strategy at the start of the hand, not at each decision point. Before the first action, list the whole range of hands you could hold at this point and what each would do on every later street. The source calls this 'reading your own hand'.
2. 2Write down each action sequence and the hand categories in it: value bets, bluffs, semi-bluffs. Any sequence containing only one category is a leak. Always raising with aces is exploitable only if aces are the only hand that raises.
3. 3Treat each category's frequency as a variable to balance. In the source's one-variable example, balance is just how close the bluff frequency is to optimal. With several hand types, balance the value bets against the bluffs together.
4. 4Test the line by maximally exploiting it. Ask what the opponent's best fixed response is (always call or always fold) and what it earns him against your mix. Plot that payoff against your bluff frequency and look for the minimum. That minimum is your balanced point. The source also uses this method to measure how far a candidate strategy is from optimal.
5. 5Accept a near-balanced mix when the exact optimum is out of reach. Real games are too complex to solve, so aim for a strategy that shares the properties of optimal play. The source notes that a miss on either side of the optimum changes expectation only slightly.
6. 6Use the balanced strategy as your default whenever you lack enough information to exploit reliably. Before you deviate, check the read: roughly 100 showdowns come from 1,000 hands, spread across positions and board textures. Deviate only when the evidence is strong.
7. 7Re-check your reads over time. Opponents adapt, improve and try to counter-exploit you, so no edge is permanent. Drop any exploitive adjustment once the opponent has moved.

The source's bluffing game has one variable: how often A bluffs. B's maximally exploitive strategy (MES) is to fold every time if A bluffs under 5%, and to call every time if A bluffs more. Against folding, B's ex-showdown result is -3x, where x is A's bluff frequency. Against calling, it is x - 0.2. The table of B's result runs: 0% bluffs → 0; 2% → -0.06; 4.8% → -0.144; 5% → -0.15; 5.2% → -0.148; 10% → -0.1; 20% → 0; 50% → 0.3; 80% → 0.6. A's optimal bluff frequency is 5%, the point that gives B the lowest result. Bluffing 4.8% or 5.2% moves the result by only about 0.006 or 0.002, so a small miss costs little. Bluffing 50% hands B +0.3, because he simply calls. The lesson is that balance is a calibrated frequency. Bluffing a lot does not make you unexploitable. You keep most of the value while giving B nothing to exploit.

• If you hold a single hand type in a given action sequence, then add other hands to it. A mix hides whether you hold the strong hand, and predictable only means exploitable when the line is pure.
• If you have no reliable read on the opponent, then play your balanced strategy. It still extracts most of the available value and leaves him no profitable counter, and his best reply is to balance too.
• If your frequency is slightly off the optimal mix, then do not panic. The loss from small deviations is minor, as in the 4.8% to 5.2% bluff range. Large deviations in either direction hand the opponent a pure call or pure fold that beats you.
• If your best 30% of hands go for value and 10% are bluffs while the optimum is 50% value and 17% bluffs, then your strategy still holds up. It gives up some value but stays balanced and hard to exploit.
• If the opponent starts countering your exploitive play, then return to the balanced baseline. Exploiting an adapted opponent often sacrifices equity for no return.

• Equating predictable with exploitable and so randomizing without calibration. The tell is bluffing at an arbitrary frequency, such as 50% when the optimum is 5%, which lets the opponent call all the time profitably.
• Treating the opponent as a fixed 'PlayStation' strategy and exploiting it permanently. The tell is that an adjustment that used to work now loses, because the opponent has adapted or improved.
• Overreading small samples. The tell is a firm player classification built on a few showdowns. In 121 UTG hands, a 10% raiser could be anywhere from about 4% to 16%, which spans very different ranges.
• Setting strategy street by street. The tell is that your flop, turn and river lines do not share a consistent hand range, so you cannot say what you could hold at this point.
• Writing off loose-aggressive players as weak. The tell is that you fold or over-adjust against someone whose play is actually strong but does not fit conventional ideas of preflop tightness.

• ▸Spot: street, position and action sequence being audited.
• ▸Value hands: which holdings bet or raise here for value, and the share of your range they make up.
• ▸Bluffs and semi-bluffs: which holdings do the same action as bluffs, and the share they make up.
• ▸Pure-line check: does any action sequence contain only one hand category? Yes or no, and why.
• ▸Opponent's maximally exploitive response: always call or always fold, and what it earns against your current mix.
• ▸Frequency sweep: the opponent's result at your current bluff frequency and slightly above and below it. Where is the minimum?
• ▸Evidence for an exploitive deviation: number of showdowns, spots observed, and age of the read.
• ▸Decision: stay balanced or deviate, with the reason, and when you will re-check.

1. 1Fix the unit first. Record results in one consistent unit per hand (the passages use BB per hand; your construct uses big bets per 100 hands). Keep the stake level and game type in every record so samples don't mix.
2. 2Estimate your per-hand standard deviation (σ) from your own results. The source's worked case uses σ = 2 BB per hand (variance 4). Use your own number, not that one.
3. 3Choose your precision target before you collect data. Example from the source: know the win rate to within 0.01 BB/hand at 95% confidence. 95% means about 2 standard errors, so the standard error must be 0.005.
4. 4Solve for sample size. The standard deviation over N hands is σ√N, so the standard error of the per-hand win rate is σ/√N. Set σ/√N = target standard error and solve for N. With σ = 2 and a target of 0.005, N = 160,000 hands.
5. 5Convert N into calendar time using your real pace. Live at about 35 hands/hour and 2,000 hours/year is 70,000 hands/year, so 160,000 hands takes a little over two years. Online at about 350 hands/hour (several tables at about 90 hands/hour each) takes roughly three months of the same hours.
6. 6Treat any win rate measured on fewer hands than N as a wide interval, not a number. Report it as a win rate plus or minus the standard error, and do not change stakes or strategy on the point estimate alone.
7. 7Re-estimate when conditions change. Opponents adapt, stakes change and games get tougher. A win rate that holds only against yesterday's fixed opponents is temporary, so start a fresh sample when the field changes.
8. 8Audit the gap between intended and actual play. Review your logged hands for spots where you played differently than you planned, since the source says this gap is often much larger than expected. Fix leaks and raise the win rate itself, not only the measurement.

The source asks how many hands are needed to know a win rate to within 0.01 BB/hand at 95% confidence, given σ = 2 BB/hand (variance 4). The standard deviation over N hands is σ√N = 2√N, so the standard error of the per-hand win rate is 2/√N. Ninety-five percent confidence is about two standard errors, so 0.01 corresponds to a standard error of 0.005. Setting 2/√N = 0.005 gives √N = 400, so N = 160,000 hands. A live professional playing 2,000 hours a year at 35 hands/hour plays 70,000 hands a year and needs a little over two years. An online player at about 350 hands/hour (four tables at roughly 90 hands/hour each) plays the same 160,000 hands in about three months. The source adds a caveat: the same per-hand win rate online could mean up to ten times the money, but online opponents often play better at comparable limits, so the win rate cannot be assumed to carry over.

• If your sample is smaller than the N needed for your target precision, then treat the win rate as an interval and make no major stake or strategy decisions from it.
• If your edge comes from a specific opponent tendency, then expect it to be temporary and plan for the opponent to adapt or counter-exploit.
• If you face stronger players, or a mix of strong and weak ones, and you cannot choose opponents (as in tournaments), then balance your strategy so it cannot be exploited rather than relying on pure exploitation.
• If you move from live to online (or the reverse), then do not assume the old per-hand win rate transfers. Re-measure it, because field strength changes.
• If a game's outcomes are comparable in size to your bankroll (large-field tournaments with payouts above about 500 buy-ins), then do not use the risk-adjusted ring-game game-selection formula.

• Judging yourself on a short sample. The tell is that your 'win rate' swings widely between monthly reviews and your confidence tracks the last few sessions.
• Believing a past edge is permanent (the 'PlayStation theory' of poker). The tell is that results decline after opponents have seen your play, and you respond by repeating the same exploits.
• Assuming the hands you intend to play are the hands you actually play. The tell is that logged hands show plays that contradict your stated strategy.
• Carrying a win rate across environments unchanged, such as live to online. The tell is that projected earnings, like the tenfold figure, never materialize because the field is tougher.
• Maximizing expected value while ignoring risk and utility. The tell is that you take on variance or stakes that threaten your bankroll in pursuit of a higher expected win rate.

• ▸Game and stake: record the game, limit and format (live/online) so samples are not mixed.
• ▸Hands played (N): the running total of hands in this sample.
• ▸Net result (in BB): total won or lost, in big blinds or big bets, for the sample.
• ▸Win rate per hand (or per 100): net result divided by N.
• ▸Observed σ per hand: your measured standard deviation per hand (the source's example uses 2 BB).
• ▸Standard error (σ/√N): the uncertainty on your win rate; the 95% interval is roughly win rate ± 2 × this value.
• ▸Target precision and required N: your desired interval half-width (e.g. 0.01 BB/hand) and the hands needed, N = (2σ / target)², which is 160,000 for the source's example.
• ▸Hands per hour and hours per year: your real pace, used to convert required N into calendar time.
• ▸Conditions changed since sample start: note changes in opponents, stakes or format that warrant starting a new sample.