Scatter Odds Calculation with Multiple Paylines ?

Hi, I am attempting to calculate the odds and ROI for a slot machine not too different from the Atkins machine. The machine in question has a scatter mechanic identical to the Atkins machine, where 3 scatter symbols anywhere in the window will pay out in free spins, so, given 1 pay line, I can easily calculate the odds of getting a scatter win in the same manner. My problem comes with people betting multiple lines. If people bet 1 line, their odds of getting a scatter is say, x, and getting a scatter will result in getting y free spins. If someone bets 2 lines, their odds of getting a scatter is still only x, and their resulting free spins is still just y. Therefore, the person would have just put in more money, but received no additional odds of free spins. So, it would appear to me as though the ROI for the machine changes depending on how many lines the person bets on (the more lines they bet, the less each bet contributes to the odds of a scatter win, and the lower the ROI). I'm curious if people would refute that, and if so, what am I missing? Also, I would like to know exactly how to factor that idea into an ROI calculation for n number of lines.
A couple notes here: my machine assumes that if the payout for 3 scatters is 5 free spins, they will still receive just 5 free spins regardless of how many lines they bet on when it hits. I'm not sure if the Atkins machine assumes that a person betting 2 lines when a scatter hits would get 2 * 5 lines?
Also, while the number of lines does not affect the number of free spins won from a scatter, it would certainly affect the ROI from those spins (as each free spin will only pay on the pay lines from the original bet).

Any ideas or contributions for this would be appreciated. Thanks,

On the Atkins game, the scatter pay is multiplied by the total bet. This way, although a player has the same scatter hit frequency, they will get paid more when they hit it.

On your game, you are awarding free games based on a scatter initiator. In this case, you should make the free games played at the same number of lines and bet per line as the initiating game. This way, the player could win free games while betting a single line, but the win in the bonus will be based on a single line. If the player wagers 15 lines, the bonus will be 15 times larger because of the increased chances of winning in the bonus. The return works out to be the same regardless of the number of lines or bet per line.

Most slot machines that award free games work this way.

Quote: cajunage

...

I think you are confusing percentages (which are relative) with credits (which are absolute).

In the case of a game like Atkins Diet, the number of paylines in play does not affect the long-term return of the game. The person playing 1 payline will, over the course of eternity, have lost the same *percentage* of their wagers as the person who plays all 20 lines. However, the 20-line player will have lost 20 times as many *credits* because they were betting 20 times as many credits as the single-line player.

Free spins are typically played using the same bet as the spin which triggered them. So if someone is playing one line and gets the free spins, the free spins will only use that one line. If all 20 lines were being played when the free spins were triggered, then the free spins will use all 20 lines.

The "motivation" to play all 20 lines comes from the likelihood that you will win more *credits* from the free spins if you are playing all 20 lines when you trigger them, since it is easier to win something when playing 20 lines than it is when playing only one line. But in the long run, the 20-line player will not win a greater *percentage* of their wagers than the single-line player.

Edit: I think CrystalMath explained it better than I did.

Thanks for the responses both of you, it helped to clarify my thinking and force me to go over assumptions I was making about certain things. I think I was just not seeing the pretty basic commutative property logic (i.e. b * n * p = n * b * p). In other words, I had already assumed the spin payouts worked just like you are both saying they should (I mentioned as much at the bottom of the original post), but I just wasn't seeing how the resulting payout math could be the same. Thanks again, much appreciated,