Calculating odds and probabilities for classic slot machine games with 3 reels with 10 symbols on each reel is simplicity itself. You simply multiply the chances of hitting the correct symbol on each real-time the chances of hitting it on the other 2 reels. The formula looks like this in fact:
1/10 X 1/10 X 1/10 = 1/1000
A brief explanation: if there are 10 possible symbols on each reel, then the chances of hitting 1 of those 10 is 1/10. The chances of hitting that same symbol on all 3 reels are the product of the chance of hitting it on each reel.
A casino makes its money by having payouts that aren’t equivalent to the true odds of hitting the jackpot. In a case where you have a 1 in 1000 chance of hitting a jackpot, if you paid out 999 to 1 or 998 to 1, the difference in the payout versus the actual odds would be your resulting profit.
But progressive jackpots are sometimes tremendous. Sometimes you can even find progressive jackpots of $1 million or more. In order to accomplish this, slot machine manufacturers had to become programmers, because realistically, a mechanical slot machine was limited to 3 or 4 reels with 20 or so symbols on each reel. A mechanical slot machine’s reels are actual strips of metal, and having more than 20 symbols on each of them made them too large to actually operate the machine.
But with a random number generator program, a casino can set up “virtual reels” with an unlimited number of symbols on each. They can also set up multiple reels. Since the odds of hitting the jackpot become much smaller, the amount of the jackpot can become much larger.
Here’s what the odds of hitting the jackpot are on a 3 reel machine with 30 symbols, and for a 5 reel machine with 30 symbols:
1/30 X 1/30 X 1/30 = 1/27,000
1/30 X 1/30 X 1/30 X 1/30 X 1/30 = 1/810,000
If the odds of hitting the jackpot are 1 in 810,000, you could easily have a payout of $500,000 or even $600,000 and still maintain a solidly profitable slot game.
Since progressive jackpots grow constantly until someone hits the jackpot, you could theoretically make a positive expectation bet on a slot machine game, IF the jackpot were large enough. In the above example, if the jackpot were $810,001, you’d be making a positive expectation bet.
The problem is that slot machine manufacturers don’t make it that easy. They use algorithms in their random number generator software that set up a % payout, so it’s unknowable to a player what the dollar amount on the jackpot must be to have a positive expectation bet.
If I only knew the settings, maybe I could wait until the jackpot were large enough to play, and make positive expectation bets all the time. But on the other hand, just having fun is usually positive expectation enough to suit me.