♠ Poker Dojo

1 · Equity — your share of the pot

Equity is the probability you end up with the best hand if the remaining cards are dealt with no further betting. It is pure combinatorics: across every way the deck can complete the board, in what fraction do you win? Split pots count as half.

Two very different quantities get called "equity", and confusing them is the most common beginner error:

• Equity vs a specific hand — exact and clean. AA vs KK is 81/19.
• Equity vs a range — the average of your equity against every hand they might hold, weighted by likelihood. This is the number that governs real decisions, because you never know their exact hand.

The coach shows both: equity vs a random hand, and vs the bot's modelled range. Watch the gap. When someone raises preflop and then bets twice, their range is strong, so your equity vs range sits well below your equity vs random. That gap is exactly what "they probably have something" means, quantified.

2 · Outs and the rule of 2 and 4

An out is an unseen card that improves you to what is probably the best hand. After the flop, 47 cards are unseen (52 minus your 2 minus the board's 3). With two cards still to come each out is worth roughly 4%; with one card to come, roughly 2%.

The rule of 4 overshoots above about 10 outs because it double-counts hitting on both streets. A better correction: outs x 4 - (outs - 8) when outs > 8.

Clean vs dirty outs. Not every out wins. Pairing an overcard on a board with a flush draw sometimes loses to the flush arriving at the same time; completing a low flush can lose to a higher one. Discount dirty outs — count 9 flush outs fully but treat 6 overcard outs as perhaps 4. The coach labels which of your outs are suspect, and comparing its outs estimate against the simulated equity vs range shows you how much the discount is worth.

3 · Pot odds — the price you are offered

When someone bets, they hand you a price. If the pot including their bet is P and you must call C, the final pot is P + C and calling breaks even exactly when:

Worked example: pot 100, they bet 50. The pot is now 150; you call 50, making 200. Required equity = 50/200 = 25%. A flush draw (35%) calls comfortably; a gutshot (16.5%) does not.

Bet size as a fraction of the pot maps directly onto the equity you need — memorise the middle column:

4 · Expected value — the only scoreboard

Every decision compares these three numbers and picks the largest. That is the entire game. Note what is not in the formula: how much you already put in this pot, how the last hand went, how you feel. Money already in the pot is not yours — it belongs to the pot, and the only question is whether the next chip you add is priced well.

Worked example. Pot 180, they bet 100, you hold a flush draw with 35% equity.

Now suppose they bet 400 instead, into the same 180:

5 · Implied and reverse implied odds

Pot odds pretend the hand ends after this call. It does not. Two corrections:

Implied odds — money you expect to win on later streets when you hit. A gutshot needing 20% but only holding 16.5% can still be a call if, when you spike the straight, you will extract another half-pot from a player who cannot see it coming. The condition is deep stacks and a disguised draw.

Reverse implied odds — money you expect to lose when you improve and are still second best. Weak top pair, a low flush, the bottom end of a straight: these hands make you feel good and then cost you your stack. The immediate math looks fine, the hand loses money over time. Whenever your "good" outcome is one someone else can beat easily, subtract value rather than add it.

6 · The decision loop — five questions, every time

Skill in poker is not knowing more facts. It is running the same short loop reliably, especially when tired, tilted, or bored. The coach panel is this loop made visible; the goal is to internalise it until the panel is redundant.

1. What do I have? Name the made hand. Name the draws and count outs. Decide which outs are clean.
2. What can they have? Start from the hands they would have played this way preflop, then delete the ones inconsistent with each action since. This is one directional filter, never a reset.
3. What is the price? Compute required equity from the bet size before you think about your cards. Doing it in this order stops you rationalising.
4. Does my equity beat the price? Compare equity vs their range to required equity. The difference, in percentage points, is your edge.
5. Which line maximises EV? Fold, call, and raise all have a number. Raising has two sources of profit — folds now and equity later. Pick the biggest, not the most comfortable.

7 · Ranges, not hands — the core mental shift

Beginners ask "what does he have?" and guess one hand. Then every card either confirms or refutes the guess and they end up wrong in expensive ways. Strong players ask "what is the set of hands he could have, and how does my hand do against that set on average?"

A range is a set of holdings with weights. Heads-up there are 1326 possible two-card combos; a preflop raise might narrow that to the top 40%, roughly 530 combos. Every subsequent action filters further. The filter only ever removes — a range never grows as the hand goes on.

How to narrow, concretely

• They bet big on a dry board — polarised: strong made hands plus a few bluffs; the medium hands are gone. Your marginal made hands are now bluff-catchers, and the question is purely whether there are enough bluffs to justify the price.
• They bet small — wide and merged: they can do this with many hands, so your equity vs range stays close to your equity vs random, and you should continue more often.
• They check — the very strong hands mostly leave the range. Now your medium hands are worth betting, and your weak ones can bluff.
• They call and call — medium strength. Calling ranges are the widest and weakest of the aggressive lines; a player who only ever calls can be bet at relentlessly.

Counting combos — the one arithmetic trick worth having

So AK is nearly three times as likely as QQ before any board cards. If the flop contains an ace, AK drops to 12 combos and AA drops to 3 — card removal quietly reshapes ranges, and it is why "he could have aces" is usually less scary than it feels.

Worked read: they raise preflop, bet the flop, bet the turn big on K♠ 9♥ 2♦ 7♣. Value hands consistent with that: KK (3 combos after a king appears... 6 minus removal), K9, K7, 99, 77, 22, AK. Bluffs: spade draws, T8, J9. If value outnumbers bluffs three to one, your ace-high needs to fold; if the board is drawy enough that bluffs are plentiful, the same hand becomes a call. Same cards, different answer, because the range changed.

8 · Polarisation and the shape of a betting range

Ranges have shapes, and the shape tells you how to respond.

• Polarised — the very best hands and the worst (bluffs), nothing in the middle. Correct size: large. Correct response: call with your bluff-catchers at the right frequency, raise rarely.
• Merged / linear — the top chunk with no gaps, including plenty of medium hands. Correct size: small to medium. Correct response: continue widely, and raise your good-but-not-great hands more.
• Condensed / capped — medium hands only, the monsters already bet. This range is defenceless against big bets; attack it.

Recognising a capped range is where most of the money is. If your opponent checks twice on a board where they would surely have bet a strong hand, their range is capped and you can bet large with almost anything — not because your cards are good, but because their range cannot fight back.

9 · Bet sizing and the mathematics of bluffing

A bet has two ways to make money: they fold now, or they call and you win at showdown. A pure bluff relies entirely on the first. The break-even fold frequency is:

A semi-bluff is far better than a pure bluff because it also wins when called and you improve. If you hold a flush draw with 35% equity when called:

This is why aggression with draws is not gambling — it is the mathematically dominant line. It is also the clearest lesson for real-world risk: take the risks that pay off in more than one way. A bet that only wins if everyone else flinches is a fragile bet; a bet that also wins on merit is robust.

Choosing a size

• Big (75–150% pot) when your range is polarised, the board is drawy, or you want to charge a draw. Big bets also grow the pot for future streets when you have the best hand.
• Small (25–40% pot) when your range is merged, the board is dry and static, or you want to deny equity cheaply and keep worse hands in.
• Consistency matters more than the number. If you only bet big with monsters and small with weak hands, your sizing becomes a broadcast. Use the same size for your strong and weak hands in a given spot.

10 · Indifference — the idea behind unexploitable play

Suppose you bet the pot on the river with a polarised range. How often should you be bluffing so that your opponent cannot beat you regardless of what they do? The answer: often enough that calling and folding are worth exactly the same to them. They become indifferent, and an indifferent opponent cannot exploit you.

This is the seed of game theory optimal play. You do not need to master it to win, but two practical habits fall out of it: always have some bluffs when you bet big, and do not fold so much that a bluff prints money against you. Beginners fail both — they only bet when strong and they fold too often when facing pressure. The two leaks feed each other.

11 · Position — information as an asset

Acting last is worth real money. In position you see their action before committing, you control the size of the pot, you can take free cards with draws, and you can bluff with better information. The same hand is materially more profitable in position than out of it — roughly the difference between a break-even holding and a clear profit.

Heads-up, the dealer posts the small blind and acts first preflop but last on every later street, which is why the dealer should play far more hands. In this trainer the position swaps every hand; notice how much more comfortable your decisions feel with the BTN tag.

12 · Variance — why a good player loses all night

Your edge per decision is small — a few percent of a pot. The noise per hand is enormous by comparison. Over one session, luck dominates completely; over tens of thousands of hands, edge dominates. The law of large numbers is the business model, and patience is the operating cost.

Concretely: a solid winning cash player earns perhaps 5 big blinds per 100 hands with a standard deviation near 90 bb per 100. The ratio means downswings of 20 to 30 buy-ins are normal, not evidence of a broken strategy. In this trainer your net can be deeply negative across 50 hands while every decision was correct — check the +EV decision counter rather than the chip count.

13 · The Kelly criterion — sizing risk to edge

Given a repeatable bet with win probability p and payout b to 1, the fraction of bankroll that maximises long-run growth is:

Three facts that make Kelly worth internalising:

• Growth is logarithmic, so ruin is infinitely bad and Kelly never risks everything. A strategy that can zero you out is not merely risky, it is disqualified.
• Betting above f* raises variance and lowers growth. At exactly 2x Kelly, expected growth is zero. Beyond that you shrink to nothing while every individual bet remains +EV. Read that sentence twice; it is the most counterintuitive result in risk.
• Kelly assumes you know your edge. You never do. Estimation error means the practical optimum is a quarter to a half of Kelly — the growth cost is small and the protection against being wrong about your own edge is large.

Go to the Risk Lab and run 4x Kelly with a genuine 60% edge. Watching a real edge get destroyed by sizing is the fastest way to feel why professionals are boring about position size.

14 · Bankroll management — Kelly in practice

Rules that matter more than the exact number: move down a stake when you dip below the threshold, never top up mid-session out of frustration, and keep the poker bankroll physically separate from money you need. Ruin is the only unrecoverable state. Everything else is a drawdown.

15 · Resulting — the habit that separates good from lucky

"Resulting" is judging a decision by its outcome. It is the most expensive cognitive error in any probabilistic domain, because it teaches you the wrong lesson at exactly the moment you are most emotional. You call correctly with 70% equity, lose, and conclude you should have folded. You call terribly with 12%, spike the river, and feel clever. Both conclusions are wrong, and the second is more dangerous.

The fix is to separate two ledgers and only take the first one seriously:

• Decision quality — given what I knew, was this the highest-EV line? Reviewable, improvable, and in your control.
• Outcome — did the cards cooperate? Noise, not in your control, and no information about decision quality in any single instance.

This is why the trainer counts +EV decisions, loose calls, and over-folds separately from your chip stack, and why the post-hand debrief reviews reasoning rather than results. Aim to make the decision counter go up and ignore the money for a hundred hands.

16 · What actually transfers to risk-taking outside poker

• Price, not certainty. "Will this work?" is the wrong question. "What does it cost, what does it pay, and how likely is it?" is the right one. Most people reject 30%-likely bets that pay 5 to 1, and take 90%-likely bets that pay 1 to 20.
• Fold cheaply, often, without shame. Folding is EV zero and always available. Most good poker is quietly declining bad prices. Optionality plus discipline beats bravado.
• Sunk cost has no place in the formula. Chips already in the pot are not yours. Neither is the time you already spent on the project.
• Size to survive. The Kelly lesson is that being right is not enough — you have to still be solvent when being right pays off. Never take a bet that can zero you, however good the odds.
• Prefer bets with two ways to win. The semi-bluff principle: a risk that pays off both if others yield and if the fundamentals land is far more robust than one that needs a single thing to go right.
• Judge process, not outcome. Keep a decision journal with your reasoning and probability estimate at the time. Review the reasoning, not the result.
• Volume is what converts edge into money. A small edge repeated often is a business. A large edge taken once is a story.

17 · Numbers worth memorising

Chen formula — a preflop score you can compute in your head

This trainer uses Chen scores to build the bot's preflop range, so the ranges you see in the coach panel are exactly this heuristic applied consistently.

18 · A starting strategy for the next hundred hands

1. Preflop: as the dealer, raise your top ~50% and fold the rest. Out of position, defend against a raise with about the top 40%, and re-raise your top 10% only.
2. Bet when you have a reason. Two valid reasons: worse hands will call (value), or better hands will fold (bluff). "I want to see what happens" is not a reason.
3. Bluff with equity. Prefer draws over air. Semi-bluffs win two ways.
4. Facing a bet, always compute the price first. Bet size to required equity, before you look at your cards again. Then compare against equity vs their range, not vs a random hand.
5. Fold the bottom of your range on later streets. If you called the flop with a weak pair and they fire again on the turn and river, their range has narrowed past you. Let it go.
6. Do not pay off the polarised line. Big bet, big raise, dry board, and you hold one medium pair: this is where beginners donate. Count their plausible bluffs before calling.
7. Attack weakness. Two checks from them means a capped range. Bet, and size up.
8. Review, do not react. Read every post-hand debrief. Track the +EV decision counter, not the chip count. Improvement shows up there first, and in the money weeks later.

This trainer plays heads-up No-Limit Hold'em with play chips only. The bot is a simple equity-and-range robot, not a real opponent — it will not exploit you, so beating it is not evidence of skill. Use it to make the loop automatic, and use the Risk Lab to make sizing instinctive.