150 free spins no deposit: an exact average, and a middle outcome nobody states

1

The average, which is exact, and the close, which is below nothing

Start with the part nobody can argue about. A hundred and fifty turns, each bought at the 10p put up by this card supposed here and promised by no cashier, commit £15.00 of stake, and 10p is the lowest price at which a turn of this reel is sold the foot of the coin steps a session hands the client. A published 96.71 per cent turns £15.00 of stake into an expected £14.51. That multiplication has three terms, all three are fixed before anybody plays, and the answer is good to the penny.

Then the condition. Forty times £14.51 is £580.40 of staking, the reel keeps 3.29 points of staking, and 3.29 points of £580.40 is £19.10. Nineteen pounds ten taken out of fourteen pounds fifty-one leaves minus £4.59, which is the arithmetic reporting that the clause asks for more than the grant was ever worth. Break-even on this reel is the reciprocal of a 3.29-point edge, which puts it at 30.4 times, and forty times stands nine and a half steps the wrong side of that.

E = B × (1 − W × (1 − RTP) ∕ g)

Reading the terms from the right: g is one, a slot counted whole against the requirement the grant’s own term; RTP is the 96.71 per cent above; W is forty; and B is the £14.51, not the £15.00, because the stake belongs to whoever granted the turns and only the winnings ever reach the reader. That substitution is the one a count of turns forces and it costs 49p on this grant before any condition is applied.

Both rows are below nothing, which is what forty times does on every published figure in this family, and £3.71 separates them. That figure is the price of a decision the terms leave open: the turns are granted on a named title and the staking is commonly allowed anywhere, so a reader who discharges it on one of the nine titles printed at 96.07 per cent has made the hole deeper by £3.71 without being told that the choice had a price.

2

What the average is made of, and how little of that is published

An average is a weighted sum of outcomes, so it has a composition, and the composition is where this page lives. Take the £14.51 apart using only figures that exist. The largest thing the reel can do on one turn is hand over 2,100 stakes the win limit, out of the session data, and the game states how often that happens: once every 3,880,481 turns. Multiply the two and the largest outcome contributes something over half a stake in every thousand turns, so across a hundred and fifty turns at 10p it comes to a penny.

One penny. That is the whole of the £14.51 whose frequency any primary source gives. The remaining £14.50 is made out of line wins, scatter awards, retriggers and multipliers whose rates are published by nobody: no base-game hit frequency exists for any title, no trigger frequency for the round exists either, and the maker’s page carries neither. So the average is a figure computed to the penny out of ingredients measured in one case out of all of them.

That is not a complaint about the maker, which publishes a return and is obliged by nobody to publish anything else. It is the reason the next three sections do not attempt a median and work with bounds instead. A quantity whose recipe is unknown can still be hemmed in, provided the hemming uses only what is on the sheet, and the sheet turns out to carry two walls.

3

A published return is a budget, and a big result has to come out of it

Here is the first wall, and it is the most useful idea on this card. A return to player is an average over turns, every outcome is a positive sum or nothing, and an average cannot be smaller than the contribution of any single outcome inside it. So for any size of result, the frequency it arrives at is limited by the return divided by the size. A result of a thousand stakes cannot arrive more often than the return divided by a thousand. A result of a million stakes cannot arrive more often than the return divided by a million. The return is a budget and a large outcome is expensive to run.

Put the family’s published figures in. The largest single symbol the maker describes is a money symbol worth 2,000 times the stake taken off the maker’s own page. Divide 96.71 per cent by two thousand and the ceiling on how often such a turn may happen is once in 2,068 turns. Not an estimate, not a reviewer’s figure, and not something the maker had to disclose: it falls straight out of the one number the maker does publish.

Carry that to the grant. A hundred and fifty turns against a rate of at most once in 2,068 gives an expected count of at most seventy-three ten-thousandths, so at most about one grant in fourteen can contain a turn returning two thousand stakes or more. Thirteen in fourteen cannot. That ceiling holds whatever the unpublished rates turn out to be, because any set of rates consistent with a 96.71 per cent return obeys it.

The same arithmetic run at smaller sizes stops biting, and it is worth saying why rather than pretending otherwise. At a hundred stakes the budget allows a turn once in about a hundred and three, and across a hundred and fifty turns that permits more than one such turn per grant, so the bound says nothing. The method only constrains the tail, and the tail is exactly the part of the distribution a grant this size is bought for.

4

Penning the middle outcome in from both sides

Two walls make a pen. The first wall is the one above: at most one grant in fourteen can hold a turn worth two thousand stakes. Suppose the budget were spent exactly that way, with one grant in fourteen collecting £200.00 at 10p and nothing of that size anywhere else. Then that one grant takes £200.00 divided by fourteen, which is £14.29 of the average, and the other thirteen share what is left: £14.51 less £14.29 is 22p. A reader in thirteen cases out of fourteen would be looking at about 22p, and the middle outcome would be 22p.

The second wall is the flat end. Suppose instead that nothing large ever happens except the win limit at its published rate, and every other turn pays the average of the rest. Then all but one grant in 25,870 receives £14.50 and that one receives £14.50 plus £210.00, so the middle outcome is £14.50 and the average exceeds it by a penny. That is the narrowest the gap can be made while still obeying the two published rates, and it is a supposition no reader should believe, because a reel whose paytable runs from two thousand down to five across one row does not pay the same thing every turn.

So the middle outcome of a hundred and fifty turns on this reel sits somewhere between about 22p and £14.50, and the average sits at £14.51 whatever happens. Both ends of that pen are built out of published figures and one stated supposition each, named in the sentence that makes it. Where inside the pen the answer actually falls depends on the frequency of every middling outcome, and that is the quantity section two found nobody states.

5

One turn that can be worth fourteen whole grants

The win limit is 2,100 stakes, which at 10p is £210.00, and the grant’s average is £14.51. One turn of this reel can therefore return a little over fourteen times what the entire hundred and fifty turns are expected to be worth. That ratio is the cleanest statement of skew available from published material, because both of its terms are published and neither of them is a frequency.

£210.00 from one turn against £14.51 for the whole grant, so a single turn of this reel can be worth fourteen grants, at 10p a turn and at the published rate of once in 3,880,481 turns the game gives for it

The rate attached to it makes the scale plain. Once every 3,880,481 turns, across a grant of a hundred and fifty, works out at roughly 25,870 grants for every one in which the limit shows up. If a hundred thousand readers each took this offer, the arithmetic expects the limit to land for about four of them, and the other ninety-nine thousand nine hundred and ninety-six would be sharing the rest of an average that has £210.00 of somebody else’s good fortune folded into it. Nobody in that position has had anything go wrong. They have received the ordinary case of a figure that is not an ordinary case.

The paytable says the same thing at a smaller scale and without any frequency at all. The top paying row of that table runs from two thousand for five of the symbol down through two hundred and fifty to five for two of it, which is four hundred to one across a single row the paytable rows in the session response. A machine that pays four hundred times as much for five of something as for two of it is a machine whose outcomes are spread, and no adjective is needed to say so.

What cannot be said is where in the spread the typical case falls, and here is the one place on this card where the section’s own subject has to be named outright. No volatility has been published for any title here at all: the word is absent from all of the maker’s pages and no session response holds a field for one. Three review sites have each put a characterisation on the original, the three do not agree, and one of them is at odds with its own table. A quantity with no publisher collects opinions, and none of those opinions is printed here as a figure.

6

The gap itself, and why its size belongs to the reel

So the distance between the average and the middle outcome falls between a penny and £14.29 on this grant, and it is no property of the offer. Nothing in the terms touches it. The count does not: doubling to three hundred turns doubles the average and leaves the shape of a turn alone. The play-through does not: forty times scales every outcome by the same factor, so it moves the average and the middle outcome together and cannot change which of them is larger. The deposit requirement does not, there being none here. What sets the distance is the paytable, the ceiling on a round and the rates behind the two, and every one of those belongs to the machine and not to the offer.

That is why this part of the domain holds the machine still and lets the offer move. Two offers with identical counts and identical conditions on two titles of one family have different gaps, because the win limits differ: 2,100 stakes on the original and as much as 25,000 on one sibling win limits, read off each title’s own data. A reader comparing offers is comparing the part that does not set the gap, and the part that does set it is printed in a data field neither the maker’s page nor any headline mentions.

One consequence of that is worth having in plain words, because it inverts the usual advice. If a reader wants the outcome they are most likely to meet to be as close as possible to the advertised average, they want a reel with a low ceiling, not a high one. The original, capped at 2,100 stakes, gives a narrower pen than a title capped at 25,000, at the same published return and on the same offer. The high ceiling is the thing the advertising is proudest of and the thing that widens the distance between the average and the case in the middle.

None of this makes an average a lie. It makes it an average, and it is the only figure in the whole arrangement that anybody has published. What a reader should not do with it is read it as a forecast of their own grant, and what this card cannot do is tell them what to read instead, beyond a pen with ends at 22p and £14.50 and the honest statement that nobody narrows it further.

7

Six minutes of turns and four hours of staking

Time is the one quantity on this card with a floor under it rather than a gap in it. On a licensed build a fresh cycle waits 2.5 seconds, automatic play is barred and every cycle is committed to singly the pace standard and the autoplay bar, so a hundred and fifty turns take at least 375 seconds. Six minutes and fifteen seconds is the grant.

The staking is a different order of thing. Five hundred and eighty pounds forty of turnover, at the 10p this card spends, is 5,804 turns, and 5,804 turns at the same floor is four hours and two minutes. So a reader is handed six minutes of play and asked for four hours of it, and the expected result of the four hours is that the six minutes are worth less than nothing: minus £4.59 against a £14.51 average that the condition more than consumes.

Those hours interact with the pen in section four in a way worth stating. A reader whose grant lands in the thin part of the distribution has a balance worth clearing and four hours of work to do on it. A reader whose grant lands in the thick part has almost nothing to clear and stops. So the four hours are not an average experience either: they are the experience of the minority whose grant produced enough to be worth the attempt, and the arithmetic of the attempt is the minus £4.59.

Both time figures are ceilings on speed and not forecasts. Not one figure on this card rests on anybody having played anything, and the only thing the pace rule settles is that a licensed reel cannot be driven past 1,440 turns in an hour, which puts £580.40 of turnover at no less than four hours whatever anybody wants.

8

What an average is good for, and what it is not

Collect it. A hundred and fifty turns at 10p commit £15.00 of stake and carry an average of £14.51, exact to the penny. Forty times that average is £580.40 of staking, costing £19.10, which leaves minus £4.59; discharged on a title printed 0.64 points lower it leaves minus £8.30, so the place the work is done comes to £3.71. Ten times, the most a British clause may reach, would have left £9.74, putting the clause’s cost at £14.33. The win limit is £210.00, or fourteen grants, at a published rate of once in 3,880,481 turns, and it puts one penny into the average. The middle outcome sits between about 22p and £14.50 and nobody narrows it.

An average is good for exactly two things here, and both of them are real. It prices the offer against another offer, because two averages computed the same way are comparable even when neither describes a typical case. And it is the only figure a condition can be applied to: the minus £4.59 is an arithmetical fact about £14.51 and forty times, and it would be the same fact if the distribution behind the £14.51 were any shape at all.

What an average is not good for is answering the question a reader actually asks, which is what will happen to them. For that the recipe is needed, and the recipe is the thing that is published in one case out of all of them. This card can do no better than a pen, and a page offering a figure for the middle outcome would be making up the most important number on it. Ends at 22p and £14.50 are what honesty leaves: wide, derived, and walled by the machine’s own published figures on both sides.

None of the above points at any particular place of play. Forty times what a grant produces is four times the ceiling a British clause may reach, so a count of this size with that condition attached comes from somewhere with no Great Britain licence to show. Away with the licence go the ten-times limit, the duty to put something about the chances of winning within reach beforehand, identity before any play at all, the pair of caps a single cycle is held to, the pause one cycle must leave before the next, the bar on automatic play, and an entry in the register of self-exclusions. The average is the same £14.51 either side of that line, and so is the pen.