math: Mills slot machine

Who doesn't love finite mathematics?  so practical, so alluring!  And awesome for gambling analysis!

I found this site that sells slot machine reels and eventually found the reels that are on my Mills Hightop.  It was on a picture that had 8 reels, but I was able to find the specific ones for me.

Here is an edited pic, and as you can see they are out of order but I drew numbers along the bottom

excuse the ordering difference

There are 20 symbols on each reel, and here is the number of occurrences:

and here are the odds of each of those coming up on each reel:

here are the outcome scenarios:

When doing this math it's important to note the NET win.  A single cherry pays two coins, but since you spent 1 coin for the spin, your net win is only 1.

The triplets (oranges / plums / bells) can also be finished by a BAR on the third reel, which adds an additional 5% chance for the winning symbol to appear on the third reel.

Interesting to note how bells are plentiful (7 of them) on the 2nd reel and 3rd reel, but on the first reel there is just a single one.

Oranges and plums only have 2 symbols each on the 2nd reel.

the third reel introduces the Lemon symbol, which is just there to mess up your triples.  (you can still win with a lemon in the third reel, if there are cherries too)

3 bars nets you a 20 coin payout plus the jackpot.  I'm told the jackpot is traditionally loaded with 80 coins, making the full payout 100.

So what is that final # in the bottom right?  That is the Expected Value of your spin.
Play 1 coin, expected to lose 0.1635 coins.  
The machine is expected to pay out 0.8365, or 83.65% of the input value.

This is an important number because the lower it gets, the less "exciting" the machine is for the player.  It just becomes a place to dump your money.  If it's too high, it cuts into the operator's profit.

These days casinos control this expected result digitally.  You can see it easily if you get a pachislo machine, do the reset, and you can change the odds there to be even more generous.  But that's boring.  I find it really interesting on mechanical machines though because they had to have this math in mind during the design.

The reels themselves can be changed!  It's not a matter of just putting on a new sticker though, as the reel has to have the physical holes that align with the symbols.  Here is an example of someone physically altering a reel to reduce the odds and increase the profit.

What happens if the jackpot hopper is empty?  If we reduce the Jackpot payout from 100 to 20 (the internal payout online) then the expected losings go from 0.1635/coin to 0.2035.
The expected payout goes from 83.65% to 79.65%