Twenty presses, an expected £1.92, and the shape that decides whether any of it arrives

1

What the £1.92 is a statement about, and what it is not

Pressed twenty times at 10p, a reel takes £2.00 of turnover, with that 10p standing as this leaf’s own assumption because a bare count carries no price alongside it. At 96 pence back in each pound, which this site holds fixed throughout and which a sibling leaf takes apart separately, those presses return an expected £1.92. Eight pence is the margin. That is the whole of the headline arithmetic and it takes one line.

The trouble begins with the word expected, which in this calculation does not mean likely and does not mean typical. It means the average across every possible run of twenty presses, each run weighted by how often it happens. A figure of that kind is exactly right about an enormous number of runs and can be a poor description of any single one. Whether it describes one run well or badly depends on how the returns are spread, and spread is a second property entirely, independent of the average.

Picture two arrangements that both average out to £1.92. In the first, most presses give back a few pence and a handful give back thirty or forty, so twenty of them produce a steady dribble and finish somewhere close to the average. In the second, nineteen presses in twenty give back nothing whatever and the twentieth occasionally gives back something very large, so twenty presses usually produce a flat nothing and occasionally produce a figure far above the average. Both arrangements honour the same rate. Only one of them makes the £1.92 a sensible thing to plan around.

Nothing on the banner distinguishes them. Nothing in the bonus document distinguishes them either, because a bonus document is written about obligations and a spread is not an obligation. The property concerned has a name in the trade and no address in any rulebook this site can cite, which puts it nowhere in writing and makes it the subject of this leaf.

One honest qualification before the modelling starts. Small counts of presses given away for nothing are a cheap thing to hand out precisely because they usually hand back little, and nobody is hiding that. What is unavailable is not the direction of travel but the size of the swing, and the size of the swing is what decides whether twenty presses are a modest amusement or a lottery ticket with a one in ten chance on it.

2

An average taken over a length nobody sits through

A rate of return is computed by whoever builds a reel, from the paytable and the frequency of each symbol, and what it describes is the limit a series of presses tends towards as the series gets arbitrarily long. The guidance behind the British disclosure standard calls the quantity an average theoretical figure, and both words are doing work: theoretical because it is derived rather than counted, average because it is a centre and not a promise.

Set that against twenty. Twenty presses is somewhere between forty and fifty seconds of pressing on a licensed build and rather less on one that is not, and it is a sample so short that the quantity governing it is not the average at all. What governs a twenty-press run is the chance that any paying event turns up inside it, and that is a different number with a different owner, published by nobody.

The mismatch can be put as a single sentence and it is the sentence this whole leaf is built on. A rate of return is a statement about the long run; twenty presses is not a long run; and the quantity converting the first into any statement about the second is precisely the quantity nothing requires anyone to publish. A reader handed £1.92 as the answer has been handed a correct figure about a question they did not ask.

How short is twenty, exactly. Part three models four reels that all return 96 pence in the pound, and on the longest-tailed of the four a reader would need two hundred presses in expectation to meet a single paying event. Twenty presses is a tenth of that. Handing somebody a tenth of the presses they would need on average to see one outcome is not a small sample, it is a fragment, and the arithmetic of a fragment is governed by whether an event happens at all rather than by any average.

This leaf observes nothing and computes everything. No run of twenty presses was performed anywhere for it, no build was opened, and nothing below is a measurement. Each figure that follows is produced by stated arithmetic from stated assumptions, and the assumptions are named in whichever sentence spends them.

3

One rate, four shapes, every one of them modelled here

To show what a spread does, four reels get modelled. Each returns 96 pence in the pound, each is pressed twenty times at 10p, and they differ only in how often a paying event turns up. The model is deliberately crude: a press either pays or it does not, and a paying press hands back a fixed quantity of stakes. Real builds pay a range of sizes. The crude version is used because it isolates the one property being examined and because this leaf refuses to invent a paytable for anybody.

Fix the hit rate and the rest follows by arithmetic. If a press pays one time in four, then over four presses one payment has to carry 0.96 of four stakes, so each payment is 3.84 stakes, which at 10p is 38.4 pence. One time in twenty makes each payment 19.2 stakes, or £1.92. One time in sixty makes it 57.6 stakes, or £5.76. One time in two hundred makes it 192 stakes, or £19.20. No judgement has been exercised anywhere in that paragraph: the payment size is forced by the hit rate once the rate of return is fixed.

Each row of the table above was built at one rate of return and one count of presses. Read the last column and the four rows split into two pairs, with the split falling between the second row and the third. Read the fourth column and the spread runs from a third of one per cent to better than nine in ten. A reader given nothing but 96 pence in the pound and a count of twenty cannot tell which pair they are standing in, and the distance between the pairs is the entire value of the offer.

4

The chance that twenty presses hand back nothing at all

The fourth column is worth deriving rather than asserting, since that column is where the whole difficulty sits. If a press pays with probability p, then it fails to pay with probability one less p, and twenty independent presses all fail with probability one less p, multiplied by itself twenty times. That is the whole calculation. Independence between presses is assumed here, and it is the one modelling assumption in this leaf that a real build might breach.

Run the four. Three quarters raised to the twentieth is 0.0032, so a reel paying one press in four hands back nothing on roughly three runs in a thousand. Nineteen twentieths raised to the twentieth is 0.3585, so a reel paying one press in twenty hands back nothing on better than a third of runs. Fifty-nine sixtieths gives 0.7145. And a hundred and ninety-nine two-hundredths raised to the twentieth is 0.9046, so the longest-tailed reel modelled here returns nothing at all on 90.46 per cent of twenty-press runs.

Two of those bars stand and two are taken away entirely, and the rate of return behind every one of them is the same 96 pence in the pound. That is the finding in a picture. An average cannot be read as a forecast for twenty presses, because the average is silent about the only question a twenty-press run turns on, which is whether a paying event lands inside it.

Say what the ninety per cent does not mean, since the direction is easy to overstate. It does not mean the reel is worse. Across the ten runs in which nine return nothing, the tenth returns £19.20, and ten runs together return £19.20 against £20.00 of turnover, which is the same 96 pence in the pound as everywhere else. Nothing has been taken away by the shape. It has been redistributed, out of the many and into the few, and a reader holding one run is nearly always among the many.

5

Where the middle run and the average part company

The split in part three’s table falls between one press in twenty and one press in sixty, and the place it falls is not arbitrary. Twenty presses at a hit rate of one in twenty contain exactly one paying event in expectation. Push the hit rate below that and the expected count of paying events inside the free presses drops under one, at which point the commonest outcome of a run becomes no paying event at all and the middle run collects nothing.

So the pivot has a formula and the formula needs only two inputs: the count of presses handed over, and the reciprocal of the hit rate. Where the count is the larger of the two, a run usually contains something and the average is a fair guide. Where the reciprocal is the larger, a run usually contains nothing and the average describes a minority of runs. Twenty presses therefore sit on a knife edge that nothing visible tells a reader which side of they are standing.

Twenty presses read as an expectation £1.92 What the middle run collects on the longest-tailed reel modelled here £0.00 The price of reading an average as a forecast £1.92

Where the difference swallows the entire figure, something unusual for this site has happened, and what produced it is the shape rather than any clause. Nothing has been deducted. No condition has been applied. The same £1.92 is simply being read two ways, once as a centre and once as a middle, and on a sufficiently long-tailed reel the two readings have nothing in common. Every other leaf here subtracts a condition from a claim. This one subtracts a second reading of the identical claim.

One in ten runs collecting £19.20 while nine collect nothing, against an advertised count that reads the same either way

The arithmetic arrives twice, as it should. By the first route, ten runs of twenty presses turn over £20.00, one of them collects £19.20 and nine collect nought, so the ten together give back £19.20, which is 96 pence in each pound staked. By the second route, the expectation per run is £19.20 multiplied by the one in ten chance of collecting it, which is £1.92, and twenty presses at 10p against £1.92 is again 96 pence in the pound. The two routes use different objects and land on the identical rate, which is what confirms the shape has taken nothing and moved everything.

6

Looking for a dispersion figure, and finding four other things

An absence is only worth printing once somebody has gone looking. The search here was for any published quantity describing how returns are spread rather than where they centre: a swing, a hit rate, a share of the return routed through a feature, anything with a second number in it. Five places were opened and each of them holds something else.

The disclosure standard comes closest and still misses. It asks that material bearing on a customer’s chances be easy to reach before commitment, and four routes lead to compliance with it: an account of the workings together with how winners get settled, a margin or an edge, a percentage return, or the probability of winning events turning up. The fourth route is the only one that touches frequency, it is one route among four joined by or, and a probability of events arriving is still not a statement about how large the arrivals are. A licensee may satisfy the whole requirement without printing a single quantity this leaf has been modelling.

The randomness standard is the second place, and what it governs is conduct rather than publication. Results must be acceptably random, and they must come out in the proportions the theory behind the build sets. A compensated reel, by which is meant one altering its odds according to what has already occurred, is forbidden. Numbers come off the generator in sequence, and throwing one away to engineer an alteration is barred as well. Each item there restricts what a reel may do. Not one of them obliges anybody to say what a reel is.

The third place is the same standard’s rule against misleading design, which forbids features reasonably expected to mislead a customer about the likelihood of particular results, and names the substitution of a losing event by a near-miss losing event as one of them. That rule is about how a shape may be dressed. It bars a build from pretending that a miss nearly was not one, and it tells a reader nothing about the shape underneath the dressing.

The fourth place is the design rule barring a reel from making a celebration out of a result worth no more than the press that bought it, with winning effects not available for such a result. Presentation again. It goes straight at one method of blurring a shape, because a reel that cheers twenty pence back on a fifty-pence press is reporting a loss as an occasion, and it publishes nothing whatever.

The fifth place is the offshore rulebook, where what is compulsory for licensed operators runs to checking age, giving access to information, self-assessment tools, watching behaviour, breaks and exclusion, limits on deposits, training of staff and principles governing marketing. Dispersion does not appear, nor does any requirement to describe how a build behaves. So the strict regime publishes a centre and no spread, and the looser regime publishes neither.

Totalled up, the published apparatus contains exactly one number about how a reel behaves, and it is the centre. Every other rule governs the process producing the outcomes or the way the outcomes are shown. A reader who wanted to know whether their twenty presses were a dribble or a lottery ticket has no published quantity to consult, inside the licence or outside it.

7

A clause laid over a balance that usually does not exist

Offers of twenty presses carry the same written conditions as larger ones, and the usual one is a multiple over whatever the presses win. Take forty, the figure an offshore bonus document reaches for when it prints one at all. Set against expected winnings of £1.92 it demands £76.80 across the reels, or 768 further presses at 10p, out of which four pence in every pound leaves an expected £3.07 behind. Subtract and the close is minus £1.15.

Check it the other way. A margin of four pence applied forty times over eats 1.6 of the winnings and runs past them by 0.6, so £1.92 multiplied by minus 0.6 is minus £1.15 again. The two arrive together, as they must, because a multiple of this size does one thing: it hands the staked money over to the margin at the rate the reel runs at.

Now put the shape back in and the clause changes character. On the longest-tailed reel modelled here, 90.46 per cent of runs produce no winnings whatever, so 90.46 per cent of runs never reach the clause at all. The expected close of minus £1.15 is an average across outcomes of which nine in ten were settled before the bonus document became relevant. For most readers the clause is not the thing that costs them anything. The shape is.

For the remaining tenth the clause is severe. A run collecting £19.20 then owes forty times that, which is £768.00 of staking, or 7,680 presses at 10p. Four pence out of every pound of that leaves an expected £30.72 behind, so £19.20 of winnings closes at an expected minus £11.52. The one run in ten that looked like the good outcome is the run in which the clause gets to do its work.

Which gives the honest summary of a twenty-press offer with forty times written over it. Nine runs in ten on a long-tailed reel end at nothing before any condition is consulted. The tenth hands over a balance and then hands it back to the margin. The advertised count is the same in both branches, the bonus document is the same in both branches, and the quantity deciding which branch a reader lands in is nowhere in writing.

8

Rules about process and dressing, and what goes with them

Counts of this size asking for no payment come from operators carrying no Great Britain licence whatever, and what deserves itemising here is the quartet of duties bearing on how results get produced and how they get shown. Not one of them would have told anybody what the spread is. Between them they decide whether the spread a reader meets is the spread a build was built with.

To start with, results must be acceptably random and must come out in the proportions the theory sets, which is the only published assurance anywhere that a build runs the shape it was designed around. Next, adaptation is forbidden: odds may not be shifted, by hand or by machine, on the strength of what has already occurred. Then the numbers driving the reel come off in sequence, with none thrown away to force a shift. Last, a losing result may not stand in as a near miss, and a result worth no more than the press that bought it may not be dressed up as a winning one.

Take those away and the four modelled reels in part three stop being four reels. Nothing then requires the hit rate to stay where it was set, nothing forbids an adjustment after a run of blanks or a run of payments, nothing prevents a sequence being sampled until it looks the way somebody wants, and nothing stops a near miss standing in for a miss so that a thin run feels like a close one. The absence of a published spread is bad enough where the spread is at least fixed. Where it is neither published nor fixed, a reader has no object to be ignorant about.

That is the sharpest form this site’s standing point takes on the present subject. The missing quantity on this leaf is unpublished everywhere, inside the licence and outside it, so on the disclosure question the licence buys a reader nothing. On the integrity question it buys them everything, because it is the only place where anybody is obliged to keep the shape honest once it has been chosen.

The concession belongs here rather than in a footnote. Not one duty in that quartet is a promise about money and not one of them improves the 96 pence in the pound. A reel bound by every one of them still returns an expected £1.92 on twenty presses and still has 90.46 per cent of runs come back empty if that is the shape it carries. What the duties provide is that the shape in front of a reader is a real property of the build rather than a thing being steered, which is a different kind of protection and worth naming accurately.

9

Twenty presses, read for what they can and cannot do

What follows is a short way of reading any small count of presses, and it produces a range rather than a figure, which is the correct output when the governing quantity is missing. It takes less time to run than reading the bonus document.

Step one, convert the count at a press value you have stated, and treat the product as turnover rather than as value. Twenty at 10p is £2.00 of turnover and that is all it is. Step two, multiply by the rate you are working at and label the rate as yours. Step three, and this is the step everybody skips, ask how many presses would be needed on average to meet one paying event, and compare that number with the count in front of you. Step four, read the comparison: count larger means the average is a fair guide, count smaller means the commonest outcome is nothing and the average belongs to a minority of runs.

Step three has no published input, which is the whole trouble, so what a reader can honestly do is run both ends. Take a generous hit rate and a thin one, work the chance of a blank run at each, and accept the pair as the answer. On twenty presses the pair this leaf computes runs from 0.32 per cent to 90.46 per cent, which is a range wide enough to decide the question entirely and narrow enough to be worth knowing.

Two habits keep this from going wrong. Never let a feeling picked up over a couple of hundred presses pass as evidence about a shape, since a couple of hundred presses is far too short a sample to tell one in sixty apart from one in two hundred. And do not treat the published rate as the answer to a question about a short run, since it is the answer to a question about an arbitrarily long one and the two questions have different answers.

The last word belongs to the offer rather than the arithmetic. Twenty presses given away for nothing cost a reader nothing and they are a real thing handed over. Somebody who wants twenty turns of a reel and intends to stop there has lost nothing by taking them. The arithmetic above is for anybody planning around the £1.92, and the conclusion for that reader is that the figure is correct, that it describes a length of play they will not reach, and that the quantity converting it into a statement about twenty presses is not published anywhere they can look.