Many slots have a scatter: a symbol that pays or triggers a feature wherever it lands, with no need to line up from the left. That makes the odds of collecting three a different sum from the one for a line or a way. This post works it through with a made-up reel. The numbers are an illustration of the method. They are not the reels, symbol counts or feature rates of any real game, including the one named at the end.
Setting up the illustration
Suppose a five-reel slot has a scatter that can appear on any reel, and that on each reel the chance of at least one scatter being in view is p = 0.2, or 1 in 5. Treat the five reels as independent: what one reel shows does not affect another. Both are simplifications, and a real game's reels may well differ. But they give a clean model, and it is the standard one.
With those assumptions, each reel is a yes-or-no trial with a 0.2 chance of yes. The number of reels showing a scatter, k, follows a binomial distribution.
The formula, and why it works
The chance of exactly k scatter reels out of 5 is
P(k) = C(5, k) × pk × (1 − p)5−k
There are three parts. pk is the chance that k chosen reels all show a scatter. (1 − p)5−k is the chance that the other reels all miss. C(5, k) counts how many different sets of reels could be the winning ones, because the scatters may sit on reels 1, 2 and 3, or 2, 4 and 5, or any other three.
For exactly three, C(5, 3) = 10, so:
P(3) = 10 × 0.23 × 0.82 = 10 × 0.008 × 0.64 = 0.0512, or 5.12%
The whole distribution
The same working for every count, with p = 0.2:
| Scatter reels | Working | Chance |
|---|---|---|
| 0 | 1 × 0.85 | 32.768% |
| 1 | 5 × 0.2 × 0.84 | 40.960% |
| 2 | 10 × 0.22 × 0.83 | 20.480% |
| 3 | 10 × 0.23 × 0.82 | 5.120% |
| 4 | 5 × 0.24 × 0.8 | 0.640% |
| 5 | 1 × 0.25 | 0.032% |
Check that the column adds up: 32.768 + 40.960 + 20.480 + 5.120 + 0.640 + 0.032 = 100.000. If a binomial table does not sum to 100%, there is an arithmetic slip somewhere. Notice also that the average is n × p = 5 × 0.2 = 1 scatter reel per spin, and the single most likely result is exactly one.
Three or more
A feature that triggers on three scatters normally triggers on four or five as well, so the figure that matters is "at least three". Add the bottom three rows:
5.120% + 0.640% + 0.032% = 5.792%
That is 1 chance in 17.3, so in this model you would expect the trigger about once every 17 spins on average. The shortcut that avoids adding long lists is to subtract the first three rows from 1: 1 − (32.768% + 40.960% + 20.480%) = 5.792%. Both routes agree, which is the checking habit worth keeping. Averages are not promises, though: why a slot is never due shows how long a dry run can last without anything being wrong.
How sensitive the answer is
The per-reel chance is the number everything hangs on, and the result is far from proportional to it. Running the same formula at other values of p:
At p = 0.1 the three-or-more chance is 0.856%, about 1 in 117. At 0.2 it is 5.792%, and doubling the per-reel chance has multiplied the trigger rate by 5.792 ÷ 0.856 = 6.8. The reason is that three things must happen at once, so the per-reel chance enters the sum cubed. Take the same trigger at p = 0.3: 10 × 0.027 × 0.49 + 5 × 0.0081 × 0.7 + 0.00243 = 0.1323 + 0.02835 + 0.00243 = 16.31%. A small change in scatter frequency changes the trigger rate a lot.
What the model leaves out
- Several scatters on one reel. The model counts reels with at least one scatter; a game counting every scatter needs a different tally.
- Unequal reels. If p differs reel by reel, the plain formula no longer applies exactly, though the idea does.
- Dependence between reels. The sum assumes independence, as in the earlier post on independent spins.
Why a scatter differs from a way
In a ways-to-win game, a symbol must appear on consecutive reels from the left, and every extra matching symbol on a reel multiplies the count, as set out in what 243 ways to win means. A scatter ignores position. Three scatter reels out of five can be reels 1, 3 and 5, with nothing between, and still count. That is why the C(5, k) term in the formula matters: it counts the many arrangements that all qualify.
The listing for Brasslight Slots: Casino Games says that landing three balls of yarn opens the Yarn Pick bonus round. It does not say whether the yarn behaves as a scatter, how often it appears, or how it is placed on the reels, so this post claims none of that, and the 0.2 above is a number chosen for arithmetic, nothing more. What the listing does confirm is the shape of the question: three of a symbol triggers a feature. Count the reels where it could appear, estimate the chance per reel, and the binomial formula gives the odds.
Checking it yourself
- Reproduce the 5.12%. Calculate 0.2 × 0.2 × 0.2 = 0.008, multiply by 0.8 × 0.8 = 0.64, then by 10. You should reach 0.0512.
- Use the complement. Add the chances of zero, one and two scatter reels, then subtract from 1. You should reach 0.05792.
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.