I have a friend I’m really fond of who believes in all kinds of conspiracy theories. He’s a nice guy, but he thinks the earth is flat.
- How Long To Roll Over 1000 Slot Money Chart
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- How Long To Roll Over 1000 Slot Money 2020
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He’s also convinced that the casino makes all its money from cheating. In fact, he told me once that the only fair game in the casino is craps, because you can’t fake the roll of the dice.
I’m not sure why he thinks dice are immune to being “fixed” while roulette wheels and playing cards aren’t. I understand how any of those devices could be fixed.
According to some more precise classifications, slot machines can be split into 5 volatility categories – high, medium-high, medium, low-medium and low. The variance of the slot machine outcomes (both online and land-based) are programmed in advance, solely depending on the RNG. Since the casinos and software providers don’t always reveal.
But casinos don’t need to fix any of these devices to make a profit.
They have a really simple means of making a profit and keeping their games completely random.
They examine the odds of winning for a game, then when they set their payout odds, they set them lower than the odds of winning.
What Are Odds and How Do They Work?
One of the ways of looking at the probability of something happening is by looking at the odds that it will or won’t happen.
Probability is just a ratio that compares the number of ways a specific event can happen with the number of ways it can’t happen.Here’s an easy example:
You flip a coin. What’s the probability that you’ll get heads as a result?
It’s 1/2, 50%, or 1 to 1.
That’s the probability expressed as a fraction, as a percentage, and as odds.
How Long To Roll Over 1000 Slot Money Chart
1 to 1 means there’s one way to lose and one way to win.
When you roll a 6-sided die, what’s the probability of rolling a 6?
There’s one way to roll a 6, and there are 5 ways to roll something that’s not a 6.
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So the odds are 5 to 1.
But there’s more to odds than that.
Odds also is a way of describing how much your bet pays off.
If you received 1 to 1 odds on guessing a coin toss, that would be called even money. You could bet $100, and if you won, you’d win $100. If you lose, you’d lose $100.
If you received 5 to 1 odds on guessing the outcome of a die roll, you’d break even over the long run. You’d win $500 every time you won, but you’d lose $100 on the 5 out of 6 times that you lost.
The way the casino makes its profit is by paying you winnings that are lower than the odds that would make a game break-even.
For example, if the casino made you risk $110 to win $100 on a coin toss, in the long run, the casino would make a profit.
50% of the time, they’d lose $100. The other 50% of the time, they’d win $110. It’s easy to see how they could make a profit doing that, right?
Or, if the casino paid off at 4 to 1 on guessing the correct outcome on a roll of a single die, they’d be making a clear profit, too, right?
5 out of 6 times, you’d lose $100. Only once out of 6 times would you win $400, which means the casino would come out ahead in the long run.
The Effect of Short Term Variance
If this is how the bets are set up, and if most people know it, why do people still gamble on casino games?
There are 2 reasons:
The first is that people are woefully uneducated about basic math in this country. Fewer casino customers than you think understand how the math behind these games works.
The second is that when you’re dealing with random events, in the short run, anything can happen, no matter how unlikely.
If you guess that a 6 will come up on the next roll of a 6-sided die, you’re PROBABLY going to lose.
But sometimes you’ll win.
In math, there’s something called the Law of Large Numbers.
This is the premise that if you repeat something random often enough, eventually, your actual results will resemble the statistically predicted results.
If you roll a 6-sided die 6 times, you might get the same number 3 or 4 times.
But if you roll a 6-sided die 6000 times, you’ll usually see a pretty even distribution for which number comes up.
This short term variation is what gamblers call luck.
That’s how some gamblers walk away from the casino winners even though the odds are against them.
How the Casino Makes Its Profit from Blackjack
Blackjack seems like a game where the casino couldn’t have an edge. In fact, it seems like a game where a smart player might get an edge over the house just by knowing how to play his cards.
After all, the odds of getting various cards are the same for both the player and the dealer.
AND the player gets a 3 to 2 payout anytime he gets a 2-card total of 21.
It’s simpler than you think, actually. The house edge in blackjack comes from the fact that the player has to play his hand before the dealer plays hers.
If your hand busts, you lose your chips immediately, before the dealer plays her hand. Even if the dealer busts, too, you’ve lost your money.
That, by itself, is enough of an advantage for the casino to make a profit.
Even if you play with perfect basic strategy, you’ll probably bust your hand around 30% of the time.
Even if the casino busts, too, you’ve already lost your bet.
The other advantage the casino has is if the dealer gets a natural, you don’t even get to play your hand. Your only hope is if you also have a natural, which is treated as a push. You don’t win any money, nor do you lose any money in that situation.
How the Casino Makes Its Profit from Craps
Craps is known for being exciting and for having a wide variety of bets you can make.
But every bet at the craps table (save one) pays off at less than the odds of winning. The only bet where that isn’t true is called (appropriately enough) the odds bet, and it pays off at true odds.
But to place that bet, you must have first placed a bet on the pass line or the don’t pass line. (Or the come or don’t come.)
Those are the best bets at the craps table, by the way, but they still give the house an edge of 1.41% or 1.36%. The other bets at the craps table are all far worse, and they all result in a profit for the casino because they don’t pay off at the same odds they have of winning.
Here’s an example of a craps bet, its odds of winning, and the payout odds for that bet:
The “any 7” bet is a single-roll bet that the results of the next roll of the dice will total 7.
The odds of winning that bet are 5 to 1.
The payout is 4 to 1.
I used the same example in the introduction when I talked about the probability of rolling a single die.
How the Casino Makes Its Profits from Roulette
Roulette is my favorite example of how probability works, especially as it relates to casino games and the house edge. It’s a simple game, too.
You have a spinning wheel with 38 possible outcomes. The pockets in this wheel are numbered 0, 00, and 1-36. The 0 and the 00 are both green. Half the numbers 1 through 36 are black, and the other half are red.
You have a variety of roulette bets available to you, but all of them share the same house edge. (Well, except for one bet, which I’ll explain here, too.)
The easiest bets, and the ones which win the most often, are the even money bets. These are the bets where if you bet $100 on them, you win $100.
An example of an even-money bet is a bet on red. Since almost half the numbers are red, you have an almost 50% chance of winning this bet.
But the 0 and the 00 make it so that the bet doesn’t pay off at the same odds as the odds of winning.
You have 38 possible results, and 18 of them are red. That’s 18/38, or 47.37%.
If you had a statistically perfect set of 38 spins, you’d win that bet 18 times, but you’d lose 20 times. You’d win $1800 on your winning spins, but you’d lose $2000 on your losing spins. Your net loss would be $200.
If you averaged that by the number of spins you made, you’d have an average loss per spin of $5.26.
All the bets on an American roulette wheel pay off at odds that would be a break-even proposition IF the 0 and the 00 weren’t on the wheel.
The bet that wins the least often in a roulette game is a single-number bet. This is a bet on a specific number, like 18, for example.
You have a 37 to 1 shot of winning this one, but when you do win it, you only get paid off at 35 to 1. The same averages apply—this bet also has a 5.26% edge.
One bet at an American roulette table has a higher house edge than that, though. It’s the 5-number bet, which is a bet that the ball will land on 0, 00, 1, 2, or 3.
That bet has an even higher house edge—7.89%.
The only correct strategy in an American roulette game is to place any bet you want as long as it’s not the 5-number bet. No strategy can overcome the inherent mathematical edge built into the game.of people come up with roulette systems based on changing the size of your bet based on what happened on previous spins of the roulette wheel. None of those systems improve your probability of winning in the long run.
The best explanation I ever read of how the various roulette betting systems work is that it’s like trying to add up negative numbers to get a positive result. No matter how you size those negative numbers, they’re still negative, and you’ll still wind up with a long-term loss.
How the Casino Makes Its Profit from Slot Machines and Video Poker
Slot machines and video poker do the same thing. They just have more possible prize amounts on their pay tables.
But each of those potential payouts has a probability.
The payout for that possibility is always lower than the probability that you’ll hit it.Let’s use a simple example:
You have a slot machine with 10 symbols, and each of those symbols has an equal probability of coming up. Let’s suppose that the top prize on the machine is for lining up 3 cherry symbols. The payout for that is 900 for 1.
What’s the probability of having that outcome?
It’s 1/10 X 1/10 X 1/10, or 1/1000.
If you make 1000 statistically perfect spins, you’ll lose $1000 (betting a dollar per spin). You’ll win $900. That’s a net loss of $100 over 1000 spins, which means that this slot machine has a 90% payback percentage and a 10% house edge.
Video poker games work the same way, although the probabilities are driven by the probabilities inherent in poker hands and decks of cards. The probability of getting a flush playing Jacks or Better video poker is the same as it would be if you were dealing out of a deck of 52 cards.
I should point out one big difference between gambling machines (like slots and video poker) and table games.
The payouts for table games are made on an X to Y basis. For example, a payout on a single-number roulette bet is 35 to 1.
This means that if you lose, you lose $1.
If you win, you keep your bet, and you win $35 on top of it.
But with a gambling machine, the payouts are made on an X for Y basis.
This means that if you lose, you lose the amount you bet.
But if you win, you get the winnings, but you don’t get your original bet back on top of it. You traded your wager FOR the winnings.
This implication is important, because you’ll find many games that offer an even-money payout on a specific hand or combination of reel symbols.
On a gambling machine, this is just a push. You haven’t really won anything. You just got the size of your bet back. If you were playing blackjack, this would be considered a push.
Why do casinos profit from random casino games?
They offer payout odds that are lower than the odds of winning.
In the short run, you can still win in a situation like that, but in the long run, the casino will always win. You can think of this disparity between the payout odds and the odds of winning as the house edge.
And you can think of the house edge as being like a tax on each bet you place. Another way to think of it is as a negative interest rate on an investment.
Playing casino games can be fun, but in the long run, playing casino games is a costly endeavor for a casino gambler.
It’s also a profitable endeavor for the casino.
- Slots Analysis
How Long To Roll Over 1000 Slot Money 2020
Hot Roll is a bonus feature added to various 3-reel slot machines by maker IGT. In the case of this analysis, it is based on the classic game Triple Double Diamond. If the player gets the Hot Roll icon on all three reels, then he will play a craps-based bonus game. In this game, the player will keep throwing two dice and winning money, until he rolls a seven, ending the bonus.
I played 284 spins of this game at the Golden Nugget on January 2, 2014. As I played, I recorded my play and then uploaded the video to YouTube. This page documents my results and attempts to reverse engineer the game to show you how it might have been programmed. For this type of game, with weighted reels, 284 spins is not enough to know exactly how IGT programmed it. What you see in this page is my best educated guess.
Hot Roll is a 20-line 3-reel slot machine.
Following are the rules for the base game.
- The player must play all 20 lines.
- The player may bet one to ten coins per line.
- The pay table for each line is as follows:
Hot Roll Pay Table
Event Pays Three triple diamonds (20th payline) 10,000 Three triple diamonds (payline 1 to 19) 1,200 Any three wilds 1,000 Three red sevens 100 Three purple sevens 80 Mixed sevens 50 Three 3-bar 30 Three 2-bar 20 Three 1-bar 10 Three cherries 10 Three mixed bars 5 Any two cherries 5 Any one cherry 2
Three double diamonds would be scored as any three diamonds.
The win of 10,000 is on a 'for one' basis, relative to the total amount bet. The pay table itself mentions a win of 100,000 coins, but this is based on a ten-coin bet. If the player bets less than 10 credits per payline, then he will win 1,200 only for three Triple Diamonds on the 20th payline.
- The Double Diamond and Triple Diamond symbols are wild and may substitute for any other symbol, except the Hot Roll or another wild, on the same pay line. Note that the wilds may not substitute for a cherry unless there is a natural cherry on the same line.
- A double diamond will double any win on that pay line. Likewise, a triple diamond will triple any win on that pay line.
- A combination of two wilds will both multiply. To be specific, two Double Diamonds will multiply a win by 4, one Double Diamond and one Triple Diamond will multiply a win by 6, and two Triple Diamonds will multiply a win by 9.
- All wins for three diamonds do not get multiplied.
- If the player gets three Hot Reel symbols anywhere on the screen, then he shall play the bonus game.
- If the player gets three Triple Diamonds on the 20th pay line, and makes a maximum bet of 200 credits, then he shall be paid 10,000 for 1 on that line, instead of the usual 1,200 for 1 for three Triple Diamonds.
- The pay lines are drawn as follows:
Hot Roll Paylines
Line Reel 1 Reel 2 Reel 3 1 middle middle middle 2 top top top 3 bottom bottom bottom 4 top middle bottom 5 bottom middle top 6 middle top middle 7 middle bottom middle 8 bottom middle bottom 9 top middle top 10 top middle middle 11 bottom middle middle 12 middle top top 13 middle bottom bottom 14 top top middle 15 bottom bottom middle 16 middle middle top 17 middle middle bottom 18 top bottom top 19 bottom top bottom 20 top top bottom
Rules for the bonus game.
- The player shall keep rolling a pair of dice until he rolls a seven.
- Per Nevada law, the outcome of each die is independent and each side has a 1/6 probability, as with real dice.
- If the player gets a seven on the first roll, then he shall win 7 times the total amount bet.
- Otherwise, the player shall win the amount in the following table. Wins are based on the total amount bet. The player will keep all wins until he rolls a bonus-ending seven.
Hot Roll Bonus
Based on my 284 spins, I put together the order of the reel stripping and count how often each reel stopped on each position. The following table shows my results.
Hot Roll Data
|Reel 1||Reel 2||Reel 3|
|Double Diamond||2||Double Diamond||4||Double Diamond||1|
|Triple Diamond||1||Triple Diamond||4||Triple Diamond||4|
|Hot Roll||28||3-bar||23||Hot Roll||22|
|2-bar||20||Hot Roll||31||Purple 7||14|
Triple Double Diamond Analysis
The way that three-reel slot machines usually work is to pick one random number for each reel and then map it to a position on the reel strips, according to how each stop on each reel is weighted. It is not unusual for the total number stops to sum to an even power of 2. I don't know the total number of stops for this game but for the sake of example I will assume it is 28 = 256.
The following table is my best estimate of the actual reel weights. I combined trying to keep the proportions the same as the actual data and trying to achieve what I felt was a believable return for the game. You may recall that in my Las Vegas penny slot survey that the Golden Nugget came in 48th place out of 71, with an average return of 90.85%.
Hot Roll Data — Hypothetical Reel Weights
|Reel 1||Reel 2||Reel 3|
|Double Diamond||2||Double Diamond||3||Double Diamond||1|
|Triple Diamond||1||Triple Diamond||3||Triple Diamond||4|
|Hot Roll||25||3-bar||21||Hot Roll||20|
|2-bar||18||Hot Roll||28||Purple 7||13|
The way the game would be programmed, based on these weights, would be to choose three random integers from 0 to 255 (programmers always start counting at zero). It would then map those numbers to a specific stop on the reel according to the following ranges for each stop. The game would then stop each reel on the predestined position in the middle row.
Hot Roll — Reel Stop Ranges
|Reel 1||Reel 2||Reel 3|
|blank||0||blank||0 to 1||blank||0|
|Double Diamond||1 to 2||Double Diamond||2 to 4||Double Diamond||1|
|blank||3 to 14||blank||5 to 6||blank||2|
|Triple Diamond||15||Triple Diamond||7 to 9||Triple Diamond||3 to 6|
|blank||16 to 20||blank||10 to 12||blank||7 to 8|
|Purple 7||21 to 25||2-bar||13 to 16||1-bar||9 to 49|
|blank||26 to 29||blank||17||blank||50 to 57|
|Cherry||30 to 34||Cherry||18 to 22||Red 7||58 to 74|
|blank||35 to 39||blank||23 to 29||blank||75 to 86|
|3-bar||40 to 46||1-bar||30 to 46||2-bar||87 to 103|
|blank||47 to 63||blank||47 to 58||blank||104 to 113|
|3-bar||64 to 88||Purple 7||59 to 77||3-bar||114 to 125|
|blank||89 to 106||blank||78 to 96||blank||126 to 136|
|Hot Roll||107 to 131||3-bar||97 to 117||Hot Roll||137 to 156|
|blank||132 to 150||blank||118 to 137||blank||157 to 170|
|2-bar||151 to 168||Hot Roll||138 to 165||Purple 7||171 to 183|
|blank||169 to 194||blank||166 to 185||blank||184 to 194|
|Red 7||195 to 218||1-bar||186 to 212||1-bar||195 to 201|
|blank||219 to 237||blank||213 to 239||blank||202 to 243|
|3-bar||238 to 246||1-bar||240 to 249||1-bar||244 to 251|
|blank||247 to 249||blank||250 to 252||blank||252 to 254|
|1-bar||250 to 255||Red 7||253 to 255||Cherry||255|
Let's look at an example. Suppose the random numbers were as follows:
- Reel 1: 222
- Reel 2: 0
- Reel 3: 175
From the above table, you can see the 222 for reel 1 gets mapped to the blank between the red 7 and the 3-bar. The 0 for reel 2 gets mapped to the first blank, above the double diamond and below the red 7. The reels wrap around so the red 7 at the bottom is above the blank at the top of the list. The 175 for reel 3 would be mapped to the purple 7. The outcome would then look as follows.
Based on the weights above, the following table shows the number of combinations of each win by its multiplier for reel 1.
Payline 1 Probability Combinations
|Three triple diamonds||1,200||12||-||-||-||-||-||12|
|Any three wilds||1,000||78||-||-||-||-||-||78|
|Three red sevens||100||1,224||1,398||1,563||180||540||351||5,256|
|Three purple sevens||80||1,235||784||822||131||363||175||3,510|
|Three mixed bars||5||406,775||23,793||32,056||-||-||-||462,624|
|Any two cherries||5||8,715||9,949||15,849||-||-||-||34,513|
|Any one cherry||2||677,010||-||-||-||-||-||677,010|
The next table shows the return combinations for each win. Each cell in the main body of the table is the product of the win, multiplier, and number of combinations from the table above. The total number of possible combinations is 2563 = 16,777,216. Dividing the total return combinations in the lower right cell of 10,717,885 by the total possible combinations of 16,777,216 we get 63.88%. So, for the one credit bet on the center payline, the player can expect to get back 0.6388 credits, not counting the bonus.
Payline 1 Return Combinations
|Three triple diamonds||1,200||14,400||-||-||-||-||-||14,400|
|Any three wilds||1,000||78,000||-||-||-||-||-||78,000|
|Three red sevens||100||122,400||279,600||468,900||72,000||324,000||315,900||1,582,800|
|Three purple sevens||80||98,800||125,440||197,280||41,920||174,240||126,000||763,680|
|Three mixed bars||5||2,033,875||237,930||480,840||-||-||-||2,752,645|
|Any two cherries||5||43,575||99,490||237,735||-||-||-||380,800|
|Any one cherry||2||1,354,020||-||-||-||-||-||1,354,020|
Since the reel stops are weighted, this analysis must be repeated for each payline. To prevent this page from getting too long for the other 19 paylines I will just present the return in the following table. Note the bottom right cell shows an average return for the base game of 68.69%.
The next table shows the number of combinations for each type of win over all 20 paylines.
Win Combinations over all 20 Paylines
|Three triple diamonds||10,000||15||-||-||-||-||-||15|
|Three triple diamonds||1,200||672||-||-||-||-||-||672|
|Any three wilds||1,000||3,045||-||-||-||-||-||3,045|
|Three red sevens||100||15,307||18,411||22,128||4,515||11,843||7,341||79,545|
|Three purple sevens||80||20,374||27,735||32,131||5,250||14,364||8,910||108,764|
|Three mixed bars||5||7,083,199||560,024||721,851||-||-||-||8,365,074|
|Any two cherries||5||185,229||296,674||371,898||-||-||-||853,801|
|Any one cherry||2||13,361,573||-||-||-||-||-||13,361,573|
The following table shows the expected number of each kind of win over all 20 paylines. The lower right cell shows the player can expect 1.46 wins per bet.
Expected wins over all 20 Paylines
|Three triple diamonds||10000||0.000001||0.000001|
|Three triple diamonds||1200||0.000040||0.000040|
|Any three wilds||1000||0.000181||0.000181|
|Three red sevens||100||0.000912||0.001097||0.001319||0.000269||0.000706||0.000438||0.004741|
|Three purple sevens||80||0.001214||0.001653||0.001915||0.000313||0.000856||0.000531||0.006483|
|Three mixed bars||5||0.422192||0.033380||0.043026||0.498597|
|Any two cherries||5||0.011041||0.017683||0.022167||0.050891|
|Any one cherry||2||0.796412||0.000000||0.000000||0.796412|
The following table shows the expected return from each kind of win over all 20 paylines. The lower right cell shows the player can expect 13.737592 credits from line pays per bet. Dividing that by a 20-unit bet, the return from the base game is 68.688%.
Expected return over all 20 Paylines
|Three triple diamonds||10,000||0.008941||0.008941|
|Three triple diamonds||1,200||0.048065||0.048065|
|Any three wilds||1,000||0.181496||0.181496|
|Three red sevens||100||0.091237||0.219476||0.395679||0.107646||0.423539||0.393802||1.631379|
|Three purple sevens||80||0.097151||0.264502||0.459638||0.100136||0.410957||0.382376||1.714759|
|Three mixed bars||5||2.110958||0.333800||0.645385||3.090143|
|Any two cherries||5||0.055203||0.176831||0.332503||0.564537|
|Any one cherry||2||1.592824||0.000000||0.000000||1.592824|
The next table shows the frequency of each win amount, after applying the multiplier, over all 20 paylines. The lower right cell shows a total win of 13.737592. Dividing this by 20, the total amount bet, results in a return for the base game of 68.688%.
Win Summary over all 20 Paylines
The rules for the bonus are stated in the rules section above. Let's start the analysis of the bonus by solving for the average win per roll, assuming it isn't a seven. The table below answers that question. The bottom right cell shows an average win of 3.733333, assuming no seven.
Hot Roll Bonus Analysis
Next, what is the average number of rolls? If the probability of an event is p then it will take on average 1/p trials for it to happen. The probability of rolling a seven is 1/6, so it takes on average six rolls to happen. However, the player doesn't win anything for the actual roll of the seven, so there are five paying rolls before the seven.
There is also a consolation prize of 7 for rolling a seven on the first roll. The value of that is (1/6) × 7 = 1.166667. So, the average win per bonus is 1.166667 + 5 × 3.733333 = 19.833333.
As a reminder, the bonus is triggered if the player gets three Hot Roll symbols anywhere on the screen. To determine the probability of it occurring on each reel, we also need to examine the blank stops immediately above and below the Hot Roll symbol that touch the center payline. For reel 1 there are, 18 (blank) + 25 (Hot Roll) + 19 (blank) = 62 stops that when touching the center payline make the Hot Roll symbol appear anywhere in reel 1, yielding a probability of 62/256 = 0.242188.
The following table shows the probability of a Hot Roll symbol appearing on each of the three reels as well as the product. The lower right cell shows a bonus probability of 1.13%.
Hot Roll Bonus Analysis
The overall return from the bonus is the probability of the bonus times the average win. This product is 0.011308 × 19.833333 = 0.224279.
After all that, we have shown the return from the base game is 68.688% and the return from the bonus is 22.428% for a total return of 91.116%. If the player bets less than 200 credits, thus losing the max coin incentive, the return drops by 0.039% to 91.077%.
I would like to emphasize that I am not claiming this is the exact return. This page is more for an exercise in slot machine design than solving for the exact return of that one game. To determine the exact return I would need to know the exact reel weights, which is information I do not have.
Video of the 288 spins this analysis is based on.
My thanks to Miplet and tringlomane for their help verifying the math above.
Written by: Michael Shackleford