One or more is not certainty: the sum, worked in pounds

Is one or more the same as certain?

One or more is not certainty. Take an event with a fixed 10% chance on each of ten independent trials, and the chance of one or more occurrences is 65.13%, while the chance of none is 34.87%.

That is arithmetic rather than a forecast, and it holds only while each trial is independent and the chance stays the same. Change either assumption and the sum changes with it.

The other half of the answer matters more at the table, because a run of losses does not make the next trial more likely to land. The Gambling Commission says so in its own player guide, and the sentence is quoted below.

Everything here is worked with £10 stakes, so you can see the money next to the maths. We ran no game, and we measured no operator.

Sources: NIST: binomial distribution; Gambling Commission: return to player; Gambling Commission: RTS 7.

Step 1 — write down the inputs

Three numbers go on the board before we can work out one or more. The first is the chance on each trial, written p, and the second is the number of trials, written n.

Then come the assumptions, said out loud. Each trial is independent, so nothing carries over from the last one, and the chance p is the same every time.

NIST’s handbook states the second one plainly: ‘The binomial distribution assumes that p is fixed for all trials.’ It gives the chance of exactly x occurrences as C(n,x) × p^x × (1 − p)^(n − x).

For this class, p is 10% and n is 10, while the stake is £10 a trial, so ten trials put £100 through the game. Those are inputs rather than predictions.

Sources: NIST: binomial distribution.

Step 2 — put the £10 through the formula

The formula has two steps. First, find the chance that nothing happens, which is (1 − p) on one trial and (1 − p)^n over n trials, and then subtract that result from one.

In words: the chance of one or more occurrences is one minus the chance of none. That is the whole of it.

Now put the numbers in, remembering that (1 − 0.10) is 0.90 and that 0.90^10 is 0.3487 to four places. One minus 0.3487 leaves 0.6513, which is 65.13%.

Try it with £10. Ten £10 stakes are £100 staked, and the sum above says nothing about what comes back. It answers one question only: did the event happen at least once?

Sources: NIST: binomial distribution.

Step 3 — read the answer beside the other answer

65.13% is not 100%, and it is not a promise about the order of results. It is the share of runs, over very many runs of ten trials, in which the event turns up one or more times.

A different sum answers a different question, because ten trials at 10% give an expected count of one occurrence, since 10 × 0.10 = 1. NIST gives that mean as np.

The two numbers travel together and are often confused, but an expected count of one does not make a single occurrence certain. The chance of none, 34.87%, is the part that gets forgotten.

Lengthen the run and the chance of one or more climbs without ever reaching certainty. At 20 trials it is 87.84%, with an expected count of two. At 50 trials it is 99.48%, and the expected count is five.

The chance of none falls in step: 34.87% at ten trials, 12.16% at 20 and 0.52% at 50. It becomes small, but it never becomes zero.

Sources: NIST: binomial distribution.

Is the next trial due?

No, it is not. The chance on the next trial is p, exactly as it was on the first.

The Commission’s player guide, read on 22 September 2026, puts it in one sentence: ‘The odds of achieving a win in the current game remain constant and are not affected by wins or losses in previous games.’

A losing run is evidence about the past rather than a lever on the future. The sum in Step 2 already allows for long empty stretches, which is why the chance of none stays above zero.

So the rule for the board is short. Count what has happened if you like, but never let the count change the number you put in for p.

Sources: Gambling Commission: return to player.

What the standards require of an online game

Online games licensed in Great Britain are held to the Commission’s remote technical standards. RTS 7 aims ‘to ensure that games and other virtual events operate fairly’, and it sets out what random has to mean.

Two of its requirements matter to this sum. The output of the generator and the game outcomes ‘should be unpredictable’, and ‘adaptive behaviour (that is, a compensated game) is not permitted’.

That is the independence assumption, written as a rule. If a remote game may not adapt to earlier results, then earlier results do not change p.

RTS 7 also bars presentation that misleads. Game designs ‘that may reasonably be expected to mislead the customer about the likelihood of particular results occurring are not permitted’, including ‘substituting losing events with near-miss losing events’.

Sources: Gambling Commission: RTS 7.

Where a compensated machine breaks the assumption

Not every machine in Great Britain works that way, because the Commission’s return to player guide describes a second kind of machine.

On a compensated machine, p is not fixed, so the one or more sum in Step 2 does not apply as written. The guide puts it this way: ‘Compensated machines are also random, but they vary the chance of winning a prize depending upon results from previous games played.’

Return to player, or RTP, is the share of the money staked that comes back as prizes over a very long run. The guide states that it ‘is an average achieved over a significant number of game plays and not each time the gaming machine is played’.

It also gives the scale of that run, since average RTP ‘is generally measured over 10,000 or 100,000 games for compensated machines and more games than this for random machines’. An evening is nowhere near that long.

Sources: Gambling Commission: return to player.

Work out one or more for your own numbers

The calculator above runs the same two lines of arithmetic, and you can do it on paper in under a minute. Here is the order for working out one or more yourself, and what the answer means when you have it.

One warning comes first. The result is only as good as the two assumptions behind it, so write them down before you write the answer.

If you cannot state p from a published rule set, the honest output is ‘unknown’, not an estimate dressed up as a number. We leave those blank rather than fill them.

Sources: NIST: binomial distribution.

What the spread looks like over ten £10 bets

Variance is how far a single run sits from the average, and it is wider than most people expect. The 65.13% covers every run in which the event lands once or more, from one occurrence to ten.

Ten £10 stakes are £100 through the game. The average hides both ends: the run with no occurrence at all, which is 34.87% of runs, and the run with several.

Nothing here says what comes back, because a chance of one or more is not a return, and this page does not model money returned.

The spread is the part that hurts, and it is the part the average hides. If a session has gone further than you meant it to, GamCare’s helpline is 0808 8020 133.

Sources: NIST: binomial distribution.

What this sum does not tell you

Independence and a constant p are requirements, not decorative small print. A game with changing conditions needs a different model, and the compensated machine above is one example.

The one or more sum says nothing about money, because it is not an expected value and it is not the operator’s share of each £1 staked. Those are different sums on different pages.

It also says nothing about a particular evening. Understanding the maths makes a game clearer, but it does not make it safer.

Next step: put your own p and n into the calculator above, then write the two assumptions beside the answer. Was this useful, or did we get something wrong? Tell us and we will correct it.

Sources: Gambling Commission: return to player.

Sources checked 22 September 2026

Three pages were opened on 22 September 2026, and each one is listed below with what it supports. The arithmetic is our own, worked from the binomial formula on the NIST page.

We did not test a game, time a session or inspect an operator, and where a figure is ours rather than a source’s, the table caption and the key facts say so.

Percentages are rounded to two decimal places, and the underlying values are 0.651322, 0.878423 and 0.994846.

Sources: NIST: binomial distribution; Gambling Commission: return to player; Gambling Commission: RTS 7.