A return figure describes twenty-five million presses, and an evening holds three hundred

1

What a return percentage is a statement about, exactly

Start where the figure is settled rather than where it is quoted. A percentage of this sort belongs to a build’s own rules screen, which is a document a reader opens before committing, so it sits in the written terms and one movement away from the lobby tile that advertised the build. Nothing about it is hidden. Almost everything about how to read it is unwritten.

The percentage reports the average theoretical proportion of staked money a build is designed to give back, taken over the full set of outcomes the mathematics contains. Three words in that sentence do the damage. Average means no individual press and no individual run is promised anything. Theoretical means the figure comes out of the specification rather than out of a log of what happened. And full set means the quantity being averaged over is the whole distribution, including the outcomes that occur once in tens of thousands of presses and carry a large share of the total.

Put money against it immediately, because the site’s arithmetic is a subtraction and subtraction needs figures. Throughout this leaf a press is worth 10p and ninety-six pence comes back in the pound, two settings this site assumes everywhere and neither of them read off any particular build. Four pence a pound therefore stays with the margin, which makes three hundred presses £30.00 of turnover against an expected £28.80 returned, at a cost of £1.20.

That £1.20 is a perfectly good figure and it is also the most over-read number in the subject. It is the mean of a distribution, and the mean of a heavily skewed distribution is a poor description of a single draw from it. A reader finishing three hundred presses with £22.00 left and a reader finishing with £41.00 left have both met a build working exactly to its specification, and neither evening says a word about whether the specification was 94 per cent or 98.

So the honest way to state what a return figure is a statement about is to state the span. It describes a population of presses large enough that the rare outcomes have appeared at their proper frequency. Part two works out how large that is on an explicit assumption about spread. Parts three to six then show how far a reader stands from it and what the distance is worth in pounds, which is the only question on this leaf that produces a figure anybody can act on.

One concession belongs here rather than at the end. The percentage is not a marketing number and it is not inflated. Where a build is operated under a British licence the specification is the specification, and a run that departs from it is a fault rather than a feature. The criticism on this leaf is not that the figure is wrong. It is that the figure is true about something a reader will never see.

2

Presses needed before an observed rate says anything

Working out how many presses settle a rate needs one quantity the specification sheet never carries: how widely a single press scatters about its own mean. Take that spread at five times the stake, which is this leaf’s own assumption, chosen because it is a modest setting for a reel-based build and labelled here so that nothing below quietly inherits it as a fact. Everything in this part is that assumption worked through, and a different assumption moves every row in proportion to its square.

On that setting the uncertainty around an observed rate shrinks with the square root of the press count, and that single relation carries the rest of this part. Take five, divide it by the square root of however many presses have been made, and out comes the spread on the observed rate. Four hundred presses leave it at a quarter of the stake, which is twenty-five points of return. To squeeze it down to the four points the margin occupies takes fifteen thousand six hundred and twenty-five presses, because five over four hundredths is a hundred and twenty-five, and a hundred and twenty-five squared is that figure.

Run the same line at tighter targets and the counts climb by the square. One point of return needs five hundred squared, which is 250,000 presses. Settling it inside a tenth of a point needs five thousand squared, which is 25,000,000. Those are not rhetorical numbers. They are the direct consequence of one square root and one assumed spread, and they are the reason a published rate is a statement about a span rather than about an afternoon.

The times in the right-hand column come from folding a published floor into this leaf’s own count. Pressing under a British licence tops out at 1,440 cycles in the hour, that being what the minimum gap between cycles leaves room for, so 15,625 presses cannot be done inside ten hours and fifty-one minutes and 25,000,000 cannot be done inside 17,361 hours. The gap is a public record and every division here is this leaf’s.

Notice which row the margin sits on. Fifteen thousand six hundred and twenty-five presses is the point at which the uncertainty around the observed rate has only just come down to the size of the thing being measured. At that count a reader can distinguish a build designed to keep four pence in the pound from one designed to keep nothing at all, and not much else. Separating 96 from 95 wants sixty-two thousand five hundred presses, and separating 96 from 95.9 wants two and a half million.

3

An evening at the reels, measured against that

Set a plain session beside the ladder. Three hundred presses is a long evening by most standards, and this leaf treats it as one, which is an assumption about habit rather than a measurement of anybody. Three hundred presses at 10p is £30.00 of turnover, an expected £28.80 back, and a spread on the observed rate of five over the square root of three hundred, which is 0.2887 of a stake, or 28.9 points of return.

Twenty-eight point nine against four. The uncertainty the evening carries is a little over seven times the whole quantity the percentage is about. Put the same thing in pounds and it stops being abstract: the expected close on the evening is minus £1.20, while one standard spread on the money is five times the square root of three hundred, at 10p, which is £8.66. The figure a reader is trying to detect is seven times smaller than the noise they are detecting it in.

£8.66 of ordinary swing on an evening whose expected cost is £1.20

Two consequences come out of that and they run in opposite directions, which is why both belong here. No evening convicts a build of anything: a session ending £20.00 down on £30.00 of turnover sits inside the ordinary range of a build working exactly as specified, so calling it tight is a conclusion the data will not bear. And no evening acquits one either, a session ending ahead saying nothing whatever about the specification.

That symmetry is worth dwelling on because the two errors are not equally common. A reader who finishes ahead rarely goes looking for an explanation. A reader who finishes behind often does, and what gets reached for is almost always something about the build rather than something about the sample. Both readings are the same mistake, committed from different directions, and the arithmetic above refuses both of them.

How many evenings would it take. Fifteen thousand six hundred and twenty-five presses at three hundred an evening is fifty-two evenings, which is one a week for a year, and at the end of it a reader holds a reading accurate to the width of the margin. Two hundred and fifty thousand presses is eight hundred and thirty-three evenings, which is sixteen years of one a week. The run the percentage describes is not long in the sense of being tiring. It is long in the sense of being a different kind of object from a life.

4

What taking the sample would cost to take

A quantity that can only be established by a sample has a price, and the price is what the sample costs to collect. This is the one place on the subject where the arithmetic produces a figure nobody expects, because the sampling is not free and is not cheap: it is paid for at four pence in every pound, which is exactly the quantity being measured.

Fifteen thousand six hundred and twenty-five presses send £1,562.50 across the reels and are expected to cost £62.50. Two hundred and fifty thousand presses send £25,000.00 across and are expected to cost £1,000.00. Twenty-five million presses send £2,500,000.00 across and are expected to cost £100,000.00. Those three figures are the price of knowing the rate to four points, to one point and to a tenth of a point respectively, and they are what a reader would have to spend to replace a published number with an observed one.

Reach the same three figures by a second route and they agree, and agreement between two routes is how every subtraction on this site gets checked. Take the turnover and multiply by the four hundredths retained: £1,562.50 gives £62.50. Or leave the turnover out altogether, take 15,625 presses, multiply by the 10p stake and then by the margin, and £62.50 appears again. Two lines sharing no step, one answer.

So the honest position on checking a published rate is this. It is checkable in principle, by anybody, using nothing but a counter and patience. It is not checkable in practice at any price a reader would pay, because the cheapest useful reading costs £62.50 and settles only whether the margin is four points or nothing, while a reading fine enough to tell one advertised build from another costs four figures. The percentage is therefore a published fact that behaves like an unpublished one.

Read the solid lengths on that drawing as the part still unsettled and the hatched lengths as the part the pressing has bought. Nothing on it is a loss in pounds. It is a ledger of ignorance being paid off, at the prices tabled in part two.

5

Two points across a year of pressing, in pounds

Having established that a reader cannot measure the rate, turn the question round and ask what the rate is worth to them. Fix a year of play at 1,500 presses a week, which is this leaf’s own setting and roughly one long evening and one short one. Fifty-two weeks gives 78,000 presses, and at 10p that is £7,800.00 crossing the reels in twelve months.

At the 96 per cent this leaf holds, £7,800.00 of turnover gives back an expected £7,488.00, a cost of £312.00 for the year. Move the rate two points down to 94 and the same turnover gives back £7,332.00, a cost of £468.00. Move it two points up to 98 and the return is £7,644.00, a cost of £156.00. The year’s pressing has not changed by a single press and the cost has moved by half as much again in one direction and a third in the other.

A year read at the published ninety-six £7,488.00 The same year read two points under it £7,332.00 What two unprinted points cost in twelve months £156.00

A hundred and fifty-six pounds arrives twice over on this leaf, from two lines that share no step. Subtract the two expected returns and £7,488.00 less £7,332.00 is £156.00. Multiply the turnover by the two points directly, and two hundredths of £7,800.00 is £156.00 as well. The two agree exactly because the rate enters the arithmetic in one place only, which is also the reason the figure scales with turnover and with nothing else whatever.

Set that beside the sampling. The difference between two builds two points apart is worth £156.00 a year to a reader pressing at this rate. Telling those two builds apart by observation takes 62,500 presses, which is ten months of this reader’s year, and the pressing itself costs £250.00 in expectation at 96 per cent. Measuring the thing is more expensive than the thing is worth, by a factor of about one and a half, and that relation holds at every level of precision because the cost of the sample grows with the square while the value of the knowledge grows linearly.

Which leaves the published figure doing work no reader can do for themselves. Reading it closely is the only route, and that means reading the version as well as the number, since a build may hold more than one specified rate and a lobby rarely says which is loaded.

6

The same two points, set against a hundred pounds

The year in part five holds the pressing fixed and lets the money move. Turn it over and hold the money fixed instead, because that is the form in which the rate compounds and the compounding is the whole reason this one figure outweighs the rest.

Money at the reels is recycled. A press returns something, the something is staked again, and the balance declines in steps of four pence per pound rather than in one subtraction. Carry that through and a balance of B at a margin of m is expected to generate B divided by m of turnover before it is gone, which is the plainest expression of compounding available on this subject. A hundred pounds at four hundredths is £2,500.00 of turnover, which is 25,000 presses at 10p.

Now move the rate. At 98 per cent the margin is two hundredths and the same hundred pounds is expected to produce £5,000.00 of turnover, which is 50,000 presses. At 94 per cent the margin is six hundredths and the hundred pounds produces £1,666.67, which is 16,667 presses. Two points up doubles the pressing a deposit buys. Two points down cuts it by a third. Nothing else a reader can alter has that shape, because every other lever acts once and this one acts on every recycled pound.

The figures in that drawing are turnover rather than loss, which matters for reading it honestly. A reader who presses until the balance is gone has lost the hundred pounds under every rate in the table. What the rate decides is how much pressing the hundred pounds bought on the way, and a build two points better hands over twice the pressing for the identical sum. Whether that is a benefit depends entirely on what somebody wanted the hundred pounds to do, and this leaf takes no view on that.

7

Levers a reader holds, and the one they do not

Three quantities decide what a year of pressing costs, and the arithmetic treats them identically: turnover is presses multiplied by stake, and cost is turnover multiplied by margin. A reader sets the first two and sets neither the third nor, in any useful sense, the choice among builds that would move it.

Halve the stake from 10p to 5p and the year’s turnover falls from £7,800.00 to £3,900.00, so the expected cost falls from £312.00 to £156.00. Halve the pressing from 1,500 a week to 750 and the identical thing happens. Both levers are real, both are entirely in a reader’s hands, and both work by buying less play. Moving the rate two points saves the same £156.00 and buys exactly as much play as before, which is the distinction worth holding on to: one lever changes the price of the thing, the other two change how much of the thing is bought.

Set out as a table the three levers look interchangeable, and they are not. The stake and the pressing are settled by whoever is playing and are settled nowhere in writing, since nothing anywhere records how often a given person presses. For the rate, the build’s own written terms settle it, in the written terms being an address a reader reaches before a penny moves. That is an odd reversal of the usual pattern on this site, where the reader’s own choices are the known quantities and the operator’s are not.

The reversal has a practical consequence. Of the three quantities, the one a reader cannot control is the one they can actually look up, and the two they control are the two they are least likely to have measured. Somebody who has never counted a week’s presses cannot name their own turnover, so the year’s expected cost is hidden from them by their own records rather than by anybody else’s drafting.

Two further levers are often reached for and neither does what it appears to do. Swapping builds between presses alters the rate from that press onward and alters nothing already spent, no build carrying any memory of the last. Stopping after a good run keeps the run and leaves the rate exactly where it was, the rate attaching to the next press and never to the balance. Both of those sit with the randomness requirement rather than with this leaf’s arithmetic.

The compounding in part six is what makes the rate the lever worth hunting for. A reader who cuts their pressing in half gets half the play. A reader who finds two points gets the same play at two thirds the cost and, on a fixed deposit, twice the pressing. Over a year on these settings the two are worth the same £156.00, and over a deposit they are not close.

8

Who is obliged to publish any of these quantities

Everything above leans on a published rate, so it is worth asking plainly who has to publish one and what the duty actually covers. In Great Britain the answer is a technical standard rather than a promise, and the standard is narrower than the argument this leaf has built on it.

A licensed build owes an account of its own rules, reachable ahead of any commitment, accurate, wide enough to cover every rule in force, and written with sensible effort put into being understood. A second strand owes the same early reach to the prizes and to the method by which the sums are arrived at. Those two obligations are why a rules screen exists, and they put the percentage in a published standard as a matter of duty while the number itself lives inside the build.

A third strand quietly carries this leaf’s long-run argument. Neither the rules, nor what pays, nor the odds on anything paying may be shifted about while a build is live, and a shift that does occur has to be set before the customer, the regulator’s guidance picking out as unacceptable a clause keeping an open-ended right to revise. Take that away and part one’s average is an average over a specification free to move under a reader mid-run, which makes 25,000,000 presses a sample of nothing.

A fourth strand is easy to pass over. What the build did, and what the customer’s own gamble did, has to reach the screen plainly and correctly and stay there long enough for somebody to take in. An outcome that cannot be read is more than an irritation. It is a press that occurred with no way for a reader to write it down, and nobody assembles part two’s sample out of presses they could not record.

Two honest qualifications attach. The first is that the disclosure strand hands a licensee four ways to discharge it and leaves the choice with the licensee: set out the mechanics and how a winner is arrived at; print the margin; print the return; or print the odds on whichever events pay. Any single one closes the obligation, so a licensed build need not carry the number this leaf works with, and where something else is carried the arithmetic above gets rebuilt from whatever was given. The second is that an offshore rulebook consulted for this site does want the conditions on a bonus stated openly and does forbid promotion pushed aggressively, so the ground outside a British licence is not bare. On randomness, on disclosure and on changes made mid-play it holds nothing whatever.

9

Three quantities the percentage quietly rests on

Three numbers have been carried through this leaf without a source, and each is marked where it is spent. Collecting them in one place is the honest close to the argument, because an assumption used ten times and admitted once is not admitted at all.

The first is the spread of a single press, taken at five times the stake in part two. Nothing obliges anybody to publish it, no primary source this site can cite states a typical value, and it is the quantity the entire sample-size ladder turns on. Raise it to eight stakes and the count needed for a four-point reading rises from 15,625 to 40,000. Lower it to three and the count falls to 5,625. The shape of the argument survives every setting, since the square-root law does not care about the constant, but the specific counts move by the square and the table says so.

The second is the press value. This site assumes ten pence everywhere and no rule fixes it, so the turnover figures and every cost drawn out of them scale straight with it. A reader pressing at 20p takes a year’s turnover to £15,600.00 and the cost of two points to £312.00, while not one row of part two moves, a count of presses being a count of presses whatever each costs. The asymmetry is worth noticing: raising the stake makes the rate matter more and makes it no easier to establish.

The third is the pace at which a reader presses. Part two converted counts into hours using the published floor for a licensed build, and that floor is a public record. Away from a British licence nothing fixes it, so the same 15,625 presses might occupy ten hours and fifty-one minutes or a fraction of that, and the hours column becomes a statement about a rule rather than about a reader. Nothing in the money arithmetic changes when the pace does. What changes is how quickly the turnover is reached, which moves the cost of a year without moving the cost of a press.

A fourth absence is not a number and deserves naming anyway. A build may be specified at more than one rate, with the loaded version a matter between the operator and the supplier, and a lobby is not obliged to say which one a reader has in front of them. Where that is so, the figure on the rules screen is accurate about the version it describes and silent about whether that is the version running, which puts the whole question back at nowhere in writing for a reader who cannot check the two against each other.

Taken together the four silences do not undermine the published figure. They fix what it can be used for. It supports the arithmetic in parts five and six, which is about turnover and compounding and needs only the rate. It does not support any statement about an evening, a week or a deposit’s actual fate, and the counts in part two are the measure of how far it falls short of that.

10

Reading a published rate without over-reading it

The procedure below is short and needs none of this leaf’s own settings, only its shape. Find the figure, convert it into a margin, multiply the margin by whatever turnover is actually in prospect, and stop there. Every error on this subject comes from carrying the figure past the point where it is a statement about turnover.

Step one, locate the number and the version it belongs to. The rules screen is the place, it is one movement from the lobby, and a build carrying a range rather than a single figure is telling a reader that more than one specification exists. Where only a house edge or a set of event probabilities is given, convert: an edge of four per cent is a return of ninety-six, and the arithmetic that follows is identical.

Step two, turn the percentage into pence. Ninety-six per cent is four pence a pound, and four pence a pound is the only form in which the figure multiplies cleanly. A reader who holds the figure as a percentage keeps reaching for the wrong operation, because ninety-six per cent sounds like a proportion of a deposit and is a proportion of turnover, and those two are a factor of twenty-five apart on a balance played through.

Step three, estimate turnover honestly rather than optimistically. Presses a week times fifty-two times the stake. On this leaf’s settings that is 78,000 presses and £7,800.00, against a deposit a reader might think of as the sum at stake. Turnover is the quantity the margin acts on and it is routinely twenty or thirty times whatever a reader would name as what they are spending.

Step four, do the two subtractions and print both. What the year costs at the published rate, what it costs two points away, and the difference. On these settings £312.00, £468.00 and £156.00. That last figure is what the published number is worth to a reader, and it is worth exactly that whether or not the reader ever manages to confirm the rate by playing, which they will not.

Two habits keep the method from drifting. Write the assumed spread down whenever a sample size is quoted, since a count of presses with no spread behind it is a number with no derivation. And keep turnover and deposit in separate columns permanently, because the single most expensive confusion on this subject is multiplying a margin by a deposit, which understates the cost by whatever factor the balance was recycled through.

What a reader is left with is narrow and genuinely useful. A published rate will not tell them how an evening goes, cannot be checked by playing at any sane price, and does predict the cost of a year’s turnover to the penny. The honest use of it is the third of those, and the honest description of the first two is that the figure was never about them.