After a long run without a bonus, it is natural to feel that one must be close. The feeling has a name, the gambler's fallacy, and it is wrong for a simple reason: a random spin does not know what the previous spins did. This post works through the numbers using a made-up feature that appears on 1 spin in 100. The figure is an illustration, not a rate taken from any particular game, and all of the maths can be checked with a calculator.
What independent means
Two events are independent when knowing the result of one tells you nothing about the other. Each spin of a random slot draws a fresh result, so the chance of the feature on spin 501 is 1 in 100 whether the previous 500 spins produced it five times, once or never.
The same point holds for the grid itself. A ways-to-win game, such as the 243-way layout described in what 243 ways to win means, shows a new set of symbols on every spin. The count of ways on one spin carries nothing over to the next.
The belief that a feature is "due" needs a counter somewhere that fills up and then pays out. If outcomes are random and independent, there is no such counter. A coin that has landed heads nine times is still 50% to land heads on the tenth.
The chance of a drought
If a feature has a 1 in 100 chance per spin, the chance of missing it on one spin is 99/100 = 0.99. Because the spins are independent, the chances of missing several in a row multiply:
P(no hit in n spins) = 0.99n
For n = 100 that is 0.99100 = 36.6%. Many people expect a 1-in-100 feature to turn up within 100 spins. It does so only 63.4% of the time, and misses more than a third of the time. A hundred spins is the average wait, not a guarantee.
| Spins | Chance of at least one hit | Chance of no hit |
|---|---|---|
| 50 | 39.5% | 60.5% |
| 69 | 50.0% | 50.0% |
| 100 | 63.4% | 36.6% |
| 200 | 86.6% | 13.4% |
| 300 | 95.1% | 4.9% |
Two things stand out. The even-money point is 69 spins, not 100: 0.9969 is just under a half, so more than half the time the first hit has arrived by spin 69. And a 200-spin drought is not rare. It happens about one time in seven and a half, and a 300-spin one about one time in 20.
The waiting time has no memory
The number of spins until the first hit follows what statisticians call a geometric distribution. Its mean is 1 ÷ p, which here is 1 ÷ 0.01 = 100 spins. Its spread is wide: the standard deviation is √(1 − p) ÷ p, about 99.5 spins, almost the same size as the mean. That is why waits of 20 spins and waits of 300 spins both turn up in ordinary play.
The shape has a property that surprises people. Suppose you have already gone 100 spins without a hit. The chance of going a further 100 without one is
0.99200 ÷ 0.99100 = 0.99100 = 36.6%
which is the same figure as for a fresh start. The 100 dry spins divide out of the calculation. This is memorylessness, and it holds for any drought length: the expected number of spins still to wait is 100 after 10 dry spins, after 100, and after 1,000. A long drought buys you no credit, and it does not cost you any either.
The expected wait for a 1-in-100 feature is 100 spins from now, every time you ask. It is never 100 minus the spins already played.
Streaks look odd and are normal
A drought of 100 spins feels like evidence that something has changed. Compare it with its mirror image: a fair coin tossed 100 times will, more often than not, contain a run of six or more heads or tails. Streaks are what independence produces. Evenly spaced hits would be the surprising result, because evenly spaced hits would mean each spin was somehow aware of the last.
What the law of large numbers does say
People sometimes reply that things must even out in the long run. They do, in proportion, but not by correction. After 1,000 spins you expect 10 hits. If the first 100 spins gave none, the remaining 900 are still expected to give 9, so the total is expected to be 9 rather than 10. The shortfall is not made up; it is simply swamped as the spin count grows. A gap of one hit out of 10 is a 10% difference after 1,000 spins and a fraction of a percent after a million.
Checking it yourself
- Use a calculator. Enter 0.99 and raise it to the power 100. You should get 0.366.
- Change the odds. For a 1-in-50 feature the same sum is 0.9850 = 36.4%. The average wait is the number in the odds, and the chance of missing it for that long is always about 37%. That 37% is 1 ÷ e from the maths of limits, and it approaches that value as the odds get longer.
The Yarn Pick bonus in Brasslight Slots: Casino Games opens when three balls of yarn land, and the 1-in-100 example above is purely hypothetical: this post makes no claim about how often anything appears in the game. What carries over is the method. Whatever the real frequency of a feature, a stretch without it does not raise the chance of it on the next spin.
For how the symbols on a single spin are counted, see how a wild changes the count on a ways-to-win slot.
Brasslight Slots: Casino Games is a cat-themed 5-reel slot with 243 ways to win, a Yarn Pick bonus, a daily prize wheel and Turbo and Auto spin. It uses virtual coins only, there is no real money in it, and coins and gems have no cash value.