Reel strips and stops: where a slot's reels land

For the odds, a slot reel is better thought of as a list than a spinning drum. This post explains that list, called a reel strip, and how a random number turns it into the symbols you see. The strip below is invented for the arithmetic. It is not a strip from Brasslight Slots or any real game, and the Play listing for that app says nothing about its strips or how it picks results, so this post makes no claim about either.

A strip is an ordered list of stops

A reel strip is the fixed sequence of symbols around one reel. Each position in the sequence is a stop. If a strip has 20 stops, they are numbered 0 to 19, and the strip is a loop: after stop 19 comes stop 0 again. What changes from spin to spin is which stop the reel rests on.

Here is an invented 20-stop strip. The symbols echo those the listing names, but their counts are made up.

Counting down the table gives 10 paws, 5 fish, 3 yarn and 2 crowns. Check the total: 10 + 5 + 3 + 2 = 20, the length of the strip.

One random number per reel

To spin, the game draws a random whole number for each reel from 0 to 19, with every stop equally likely, and the reel lands there. Because each reel gets its own draw, the reels are independent: where reel 1 stops tells you nothing about reel 2. This is the same independence that why a slot is never due relies on.

Real games may weight stops, use longer strips or give each reel its own strip. This post assumes equal chance for every stop and the same strip on every reel, to keep the sums short.

The visible window shows neighbours

On a 3-row reel, the stop that is picked sits in the middle row. The window also shows the stop before it above and the stop after it below. Because the strip is a loop, the neighbours of stop 0 are stop 19 and stop 1.

• Picked stop 1: the window reads Paw (stop 0), Crown (stop 1), Paw (stop 2).
• Picked stop 0: the window reads Paw (stop 19), Paw (stop 0), Crown (stop 1).

The first two give the same crown from different stops: in one it is in the middle row, in the other it is the bottom row. A symbol's neighbours are fixed by the strip, so what sits above and below a crown is set when the strip is built. Only the choice of stop is random.

The probability of a symbol

With equal chance for every stop, the chance that the picked stop holds a given symbol is the number of stops with that symbol divided by the strip length:

• Paw: 10 ÷ 20 = 0.5, or 50%
• Fish: 5 ÷ 20 = 0.25, or 25%
• Yarn: 3 ÷ 20 = 0.15, or 15%
• Crown: 2 ÷ 20 = 0.1, or 10%

These add to 100%, a quick check on the count.

Seen anywhere in the window versus picked

The 10% above is the chance the crown is on the picked, middle stop. The window is wider, so the chance of seeing a crown somewhere on the reel is higher. A crown at stop 1 appears in the window when stop 0, 1 or 2 is picked, since each of those windows includes stop 1. The second crown, at stop 11, appears when stop 10, 11 or 12 is picked. These two sets of three do not overlap, so the crown is visible on 3 + 3 = 6 of the 20 stops: 6 ÷ 20 = 30%. Yarn works the same way: stops 5, 9 and 15 have separate windows, so 9 of 20 stops show yarn, which is 45%. Counting windows is the only safe method, because windows that overlap, as the fish at stops 16 and 18 do, are easy to double count.

Five reels multiply

Chances multiply across independent reels. The chance that all five picked stops hold a crown is 0.1 × 0.1 × 0.1 × 0.1 × 0.1 = 0.1⁵ = 0.00001, which is 1 in 100,000. Three crowns on the first three reels is 0.1³ = 0.001, or 1 in 1,000. A different way to reach the same answer: five reels of 20 stops give 20⁵ = 3,200,000 equally likely combinations of stops, and the number of those with a crown on every reel is 2⁵ = 32. Then 32 ÷ 3,200,000 = 0.00001, matching.

That is the picture of a spin: one combination chosen from millions. The companion posts on what 243 ways to win means and hit frequency use these chances to count routes and to work out how often anything pays.

What this does not tell you

Two games with the same symbols can have quite different strips, so the look of a reel says little about its odds. The Play listing for Brasslight Slots: Casino Games names the symbols and the 5 reels with 3 rows, but gives no strip lengths, counts or stop weights. The figures here show the method, and they are not statements about the app.

Checking it yourself

1. Count and divide. Tally each symbol; the tallies must add to 20. Crown: 2 ÷ 20 = 10%.
2. Slide the window. For each stop 0 to 19, list it and its two neighbours, wrapping at the ends. Count the windows with a crown: you should reach 6, or 30%.
3. Multiply across reels. 0.1⁵ should give 0.00001.