[Math] How to one calculate the probability and payout for a slot-machine that has multiple match lines

probability

After reading this answer on gamedev.stackexchange, and this answer here on math.stackexchange, I totally understand how to calculate the payout for a slot machine.

To summarize: In order to pay-out 95% of every dollar spent on a three wheel slot machine, (each wheel has 10 unique symbols) a jackpot match of 10-10-10 would pay out $950.

Where everything breaks down for me is the following.

Say you have a slot machine just like above, with three wheels. However, for every dollar spent (up to five dollars), you light up a new "row" that can provide new sets of "matches" per spin.

Imagine a 3×3 grid, made up of 3 wheels, like so:

                   Diagonal
   \ __  __  __ / 
    |__||__||__| - top row
    |__||__||__| - middle row
    |__||__||__| - bottom row
   /            \
                   Diagonal

So for each spin:

  1. Paying one dollar you could only match three across the middle.
  2. Two dollars could match the middle row and the bottom row.
  3. Three dollars could match the top, middle and bottom rows.
  4. Four dollars could match the same as three dollars, plus a diagonal line from the top left symbol to the bottom right symbol.
  5. Five could match the same as four, plus a diagonal line from the bottom left symbol to the top right symbol.

How does this affect the probability / odds, and more importantly, how should it affect the payout to keep the machine paying 95 cents per dollar spent?

Best Answer

The concept that you need to treat this very simply is linearity of expectation. That means that the expectation values for different random variables simply add up, even if the variables are correlated. Thus, you don't have to work out the rather complicated joint probabilities for one or more of the rows simultaneously showing a match; you can just treat each row separately, and the expectation value will be the sum of the expectation values for the individual rows. In your case, that means that you don't have to change the payout at all. Since each dollar buys an additional row and each row adds the same amount to the total expectation value as the one row you already know how to calculate, the payout should be just the same as before. (Specifically, it should be $\$950$ per match; if several rows match simultaneously, the payout should be $\$950$ for each matching row.)

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