Tommy, who is one of my blog readers asked how to calculate payout percentage of any given slot machine after reading this slot strategies article. I started writing a reply to his question in the comment, but the answer turned out much longer than I had initially planned so I decided to dedicate the answer a full post I believe it deserves.
Please note that the payout percentage can only be calculated this way if each spin is completely random. The only way you can be sure it’s completely random is if you play in a casino, which is licensed by a jurisdiction dictating the outcome of each slot spin must NOT be predetermined, it has to be completely random. Casinos bound by this restriction are all legally operating casinos in Las Vegas and Atlantic City as well as most licensed online casinos.
In order to calculate payout percentage on a slot machine with completely random RNG (Random Number Generator), you need to know 2 things beforehand: occurrence of all symbols on each reel and paytable. In a very simplistic example suppose we have a slot machine with 3 reels and with only 2 symbols
Total number of symbols on each reel is 3: 2xBAR and 1xSeven
Payout table shows 2 winning combinations for each $1 bet as follows:
First we need calculate the probability of each combination:
For 3xBAR the probability is 2/3 * 2/3 * 2/3 = 0.3
For 3xSeven the probability is 1/3 * 1/3 * 1/3 = 0.037
Note that the probability to get BAR on 1 reel is 2 out of 3 times, hence 2/3 in the equation and for Seven it’s 1 out of 3 (1/3)
We multiplied probabilities 3 times because the combination is made up from 3 times the symbol.
Now we can calculate the payout of each combination:
For 3xBAR the payout is $1 * 0.3 = $0.3
For 3xSeven the payout is $15 * 0.037 = $0.56
Finally we add the payouts of each combination together to get the total payout:
$0.3 + $0.56 = $0.86
Since we payout we just calculated is for $1 nomination (according to the paytable) then the payout for this nomination is $0.86 (each time we bet $1, on average we win back $0.86) or in other words the payout for $1 nomination is 86% (without bonus game or progressive jackpot)
Note that payout percentage can vary depending on the nomination. Higher nominations usually offer better payback.
If the slot game has a bonus round, we also need to take that into account and start by calculating the probability of getting into the bonus round. Then multiply it with the average amount you can win at the bonus round and add the result to the total payout.
Suppose in our example 3xSeven puts us into a bonus round where we are given a choice of 3 pots of gold. If we choose the right one, we win additional $3
On average we would win 1 out of 3 times in the bonus round. Since the win amount is $3, then the total amount we would win on average is 1/3 * $3 = $1 each time we get to the bonus round.
Since we have already calculated the probability to get to the bonus round (3xSeven = 0.037), then we can add another $1 * 0.037 = $0.037 to the total payout which now stands at $0.86 + $0.037 = $0.90. Now that we have taken the bonus round into account, the total payout is 90%.
If it’s a progressive slot, we need to multiply the probability to hit the jackpot with the jackpot amount and add the result to the total payout as well.
Suppose the probability to hit the progressive is one in a million and the jackpot is $10 000
1/1 000 000 * $10 000 = $0.01
After adding it to the total payout, we get $0.9 + $0.01 = $0.91 or 91% payout.
Suppose the progressive jackpot is $100 000 instead
1/1 000 000 * $100 000 = $0.1
After adding it to the total payout, we get $0.9 + $0.1 = $1 or in other words 100% payout.
If you were an advantage gambler who has done the homework like we just did here, you would know that whenever the progressive on this slot machine reaches $100 000, the total payout percentage reaches 100%, at which point playing on that slot machines becomes profitable in the long run. ‘A long run’ in this case would probably mean a few million spins, but you would be guaranteed to make profit if you had a sufficient bankroll and a lot of patience (or an army of apes pulling levers).
As you can see the math behind calculating the slot machine payout is quite simple. The most difficult part of the whole process is probably finding the information regarding the occurrence of each symbol on the reel as casinos don’t tend to publish this simply because an average gambler has no need for this type of information.
If you have any questions other than ‘Where can I recruit an army of apes’, then feel free to post them as comments and I’ll try answer them as best as I can.
UPDATE: Tommy asked how to calculate the keno video slot payout percentage, so I figured I’d throw it in here as well.
I have to admit I’m not a big fan of keno video slotmachines myself, but I believe I have a fairly good understanding how they work.
Suppose we mark 3 spots. First we need to calculate the total number of ways possible to draw 3 balls out of 80.
80/1* 79/2 * 78/3 = 82 160
Notice we substract 1 after each draw. That’s because the ball we just drew cannot be drawn again, it’s now out of the “bubble”.
Since the game draws 20 winning balls, we have to calculate the total number of ways 3 spots can be amongst the winning 20.
20/1 * 19/2 * 18/3 = 1140
The probability to hit all 3 spots is therefore 1140/82160 = 0.0139 or 1.39%
Suppose the paytable shows $40 prize for hitting all 3 on a $1 bet. The payout for hitting 3 spots out of 3 is then $40 * 0.0139 = $0.56
Suppose paytable is generous enough to award us $2 should we hit 2 spots out of 3 with every $1 bet.
The total number of ways to hit 2 spots out of the winning 20 is 20/1 * 19/2 = 190 and the total number of ways possible, that 1 ball remains amongst the 60 that don’t get drawn, is 60/1 = 60. This means there are a total of 190*60 = 11 400 different possible ways to draw 2 winning balls and 1 losing ball.
We already calculated there are 82 160 possible ways to draw 3 balls, therefore probability to draw 2 winning + 1 losing ball is 11400/82160 = 0.14
This adds another $2 * 0.14 = $0.28 to the total payout which is now at $0.56 + $0.28 = $0.84. That means the total payout is 84% for $1 card played with 3 spots.
If the paytable is the same for both 1- and 20-card version, the payout percentage remains unaffected so whether you play 1-card keno 20 times or 20-card keno once, it doesn’t really matter.