Is the question "is there a formula you can use to consistently profit at roulette?" NP complete

It is P.

This is a problem with a pretty easy to calculate solution.

Correct!

Stay away from the Roulette Wheel...

...and you will constantly profit from doing so.

Otherwise, over the long haul, the house is going to take your money at a rate of between 5 and 7 percent.

Improving Your Odds at Roulette
By Kevin Blackwood from Casino Gambling For Dummies


Remember to stay realistic about the long odds on this popular game of chance. Approach roulette with the sober realization that, with a house advantage of 5.26 percent on the American wheel, roulette is among the worst bets in a casino. Despite the odds, you can still use some simple strategies to stretch your roulette bankroll and enjoy the thrill of the spin. This article contains a few tips that can help you improve your chances of winning.

Roulette is a drain on your wallet simply because the game doesn't pay what the bets are worth. With 38 numbers (1 to 36, plus 0 and 00), the true odds of hitting a single number on a straight-up bet are 37 to 1, but the house pays only 35 to 1 if you win! Ditto the payouts on the combination bets. This discrepancy is where the house gets its huge edge in roulette.
Starting with the basics

Strategy is critical if you want to increase your odds of winning. The first time you play roulette, the players sprinkling the layout with chips may look as if they're heaping pepperoni slices on a pizza. You can make many different bets as long as you stay within the table's maximum limits. Consequently, few players make just one bet at a time.

Of course, the more bets you make, the more complicated and challenging it is to follow all the action. Here are two possible plans of attack to simplify matters:

Stick to the table minimum and play only the outside bets. For example, bet on either red or black for each spin. This type of outside bet pays 1 to 1 and covers 18 of the 38 possible combinations.

Place two bets of equal amounts on two outside bets: one bet on an even-money play and the other on a column or dozen that pays 2 to 1. For example, place one bet on black and one bet on Column Three, which has eight red numbers. That way, you have 26 numbers to hit, 4 of which you cover twice. You can also make a bet on red and pair it with a bet on Column Two, which has eight black numbers. Again, you cover 26 numbers, and 4 of them have two ways to win. Pairing a bet on either red or black with Column One (or on one of the three dozens) covers 24 numbers, and 6 numbers have two ways of winning. Spreading bets like this won't make you rich, but it does keep things interesting at the table.

Playing a European wheel

If you happen to find a single-zero European wheel, you greatly improve your odds: The house edge is half that of roulette with the American wheel — only 2.63 percent. You may see a European wheel at one of the posh Vegas casinos, such as Bellagio, Mirage, or Caesars Palace. If you can't find one on the floor, it's probably tucked away in the high-limit area along with the baccarat tables, so you may need to ask. You can also find the single-zero wheel at some other upscale casinos around the country.

Because casinos set aside the European wheel for high rollers, you're likely to find a higher table minimum, say $25. But because the house edge is half that of a double-zero wheel, the European wheel is the better roulette game to play for bigger bettors.

Your chances of winning get even better if the casino offers an advantageous rule called en prison. Sometimes available on the European wheel, the en prison rule lowers the house edge even further to a reasonable 1.35 percent. The rule applies to even-money bets. For example, say you have a $10 bet riding on black. If the ball lands on zero, your even-money bet doesn't win or lose but remains locked up for one more spin. If the ball lands on black on the next spin, the house returns your original bet of $10, but you don't win anything. If the ball lands on red, you lose. And if the ball repeats the zero number again, your bet stays imprisoned for another round.

Top 10 Best Casino Bets

By Bill Burton

http://casinogambling.about.com/od/oddsandends/tp/10best.htm

The best casino games to play are the ones with the lowest house edge or ones where the player has an advantage. Some bets are better than others. Here are the 10 best bets and games to play in the casino.
1. Casino Poker
Poker is a game of skill which is a big factor in determining the winner. The players are competing against each other and not against the house. There is no house edge except for a small amount that the casino takes out of each pot. Leaning how to play the game properly will give you a big advantage over less knowledgeable players.
More: How to Play Texas Holdem
2. Blackjack Card Counting
Learning to count cards can give the player a 1-2 percent edge over the casino. Card counting is a skill that takes practice to master. It is not illegal but counters must disguise there skills or they can be barred from playing. In New Jersey you can’t be barred but they can force you to flat bet or shuffle up on you if you are caught.
More: How to Play Blackjack
3. Video Poker
Video poker is another game of skill that has a small house edge if you make the correct playing decisions. Some video poker games offer over 100 percent payback to players utilizing perfect mathematical strategy. Many machines have a small house of less than ½ of a percent. The house edge is determined by the pay table and a player can determine the payback of the machine by reading the schedule.
More: Vicdeo Poker 101
4. Blackjack Basic Strategy
Playing Blackjack basic strategy will lower the house edge to less than one percent. The house edge is based on the number of decks used and the rules of the game. A player can determine the house edge from this information. With basic strategy you are basing your decisions to hit split or stand on set rules based the math of the game.
More: Blackjack Basic Strategy
5. Craps Don’t Pass/Don’t Come
The Don’t Pass and Don’t come in the game of craps is just slightly better than the pass line bet. The house edge is 1.14 percent. You can lower this to .59 percent by laying double odds. Most players however like to bet with the shooter on the passline.
More: How to Play Craps
6. Craps Passline/Come
The Passline and Come bets in the game of craps offer a low house edge of 1.41 percent. You can lower it to less than one percent (.61) if you take double odds on your bet.
7. Pai Gow Banker
When playing Pai Gow a player may choose to act as the Banker during the game. When you bank the game you pay a five percent commission to the house. You win the bet of any player you beat. Pai Gow requires skill in setting of the hands but a skilled player can gain the advantage over a less knowledgeable player.
8. Baccarat Banker
The banker bet in Baccarat has a smaller house edge than the player bet but it requires you to pay a commission when you win. The house edge is 1.06 percent for the Banker bet.
9. Baccarat Player
The player bet is also a good bet with a house edge of only 1.24 percent. The Baccarat bet a player wants to avoid is the tie bet which is one of the worst bets you can make.
10. Craps Place 6 and 8
Placing the six and eight at the craps table is one of the best bets in the game. The house edge is only 1.52 percent. The 6 and 8 are rolled more frequently than any number other than the 7.
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Thanks so much for your answer to the question. Just what I wanted to establish. Now could you please explain why it's a P, not NP problem? I know my questions are basic but I'm just trying to grasp a concept so any help you can give would be much appreciated.

Is this homework?

P means that it is a problem that can be solved in polynomial time.

This is a simple problem, you calculate the effective value of the roulette (a very simple function). If the effective value is greater than zero, you make money over many trials. If the effective value is below zero you lose money over many trials.

This can be solved by plugging numbers into a polynomial function. This is the simplest kind of problem.

Are you trying to understand something about roulette? Or is this question about Complexity theory?

Here's your formula in the form of a word problem. Sheeesh.

Strategy is critical if you want to increase your odds of winning. The first time you play roulette, the players sprinkling the layout with chips may look as if they're heaping pepperoni slices on a pizza. You can make many different bets as long as you stay within the table's maximum limits. Consequently, few players make just one bet at a time.

Of course, the more bets you make, the more complicated and challenging it is to follow all the action. Here are two possible plans of attack to simplify matters:

Stick to the table minimum and play only the outside bets. For example, bet on either red or black for each spin. This type of outside bet pays 1 to 1 and covers 18 of the 38 possible combinations.

Place two bets of equal amounts on two outside bets: one bet on an even-money play and the other on a column or dozen that pays 2 to 1. For example, place one bet on black and one bet on Column Three, which has eight red numbers. That way, you have 26 numbers to hit, 4 of which you cover twice. You can also make a bet on red and pair it with a bet on Column Two, which has eight black numbers. Again, you cover 26 numbers, and 4 of them have two ways to win. Pairing a bet on either red or black with Column One (or on one of the three dozens) covers 24 numbers, and 6 numbers have two ways of winning. Spreading bets like this won't make you rich, but it does keep things interesting at the table.

Playing a European wheel

If you happen to find a single-zero European wheel, you greatly improve your odds: The house edge is half that of roulette with the American wheel — only 2.63 percent. You may see a European wheel at one of the posh Vegas casinos, such as Bellagio, Mirage, or Caesars Palace. If you can't find one on the floor, it's probably tucked away in the high-limit area along with the baccarat tables, so you may need to ask. You can also find the single-zero wheel at some other upscale casinos around the country.

Because casinos set aside the European wheel for high rollers, you're likely to find a higher table minimum, say $25. But because the house edge is half that of a double-zero wheel, the European wheel is the better roulette game to play for bigger bettors.

Your chances of winning get even better if the casino offers an advantageous rule called en prison. Sometimes available on the European wheel, the en prison rule lowers the house edge even further to a reasonable 1.35 percent. The rule applies to even-money bets. For example, say you have a $10 bet riding on black. If the ball lands on zero, your even-money bet doesn't win or lose but remains locked up for one more spin. If the ball lands on black on the next spin, the house returns your original bet of $10, but you don't win anything. If the ball lands on red, you lose. And if the ball repeats the zero number again, your bet stays imprisoned for another round.

Sort of homework. Personal research..
I'm not sure which subject heading you would place it under.
Maybe it would help if I offered some context.
I don't know how much you know if the gme of roulette but in a nutshell, its a game of chance/randomness that has been around for over 300 years. It is claimed that it is unbeatable by math or any other means.
Therefore the possiblity of playing x number of games and winning a definate and consistent profit is either incalculable or does not exist.. wouldnt that then make the problem not just P but NP..?

The first case you propose is a Decision problem. The phrase you are using is "continuous and consistent profit". This implies (to me at least) a infinite number of trials. In this case it is certainly calculable. In another other case it is a nonsense question.

The probability of playing x number of games and winning a profit is also easy to calculate.

Any of these problems is P... in fact I can give you the function for any given roulette wheel if you would like.

I don't know nuffing bout maths but I do know the best odds you can get at roulette are fractionally less than Evens (50-50).
In other words no matter how you bet, the house will always have an advantage

Because the game is based on randomness you can never know for certain if you will profit from the next bet you make because the number drawn in order to give you that knowledge is not yet known. So if you lose the bet on spin x1, in order to know for certain that you will profit over x spins is to regain what you lost in spin x1 in a further spin x2. This means that the initial value of x is not finite.. ad therefore cannot be known..
Am I right in my reasoning so far?

That is correct so far (except for your use of the phrase "cannot be known". Actually it can be known, if the word you are using is "certain", then X is known to be infinite spins.)...

... but this has nothing to do with complexity theory (i.e. P or NP).

ok we're eye to eye on the infinate being a certainty. am i also right that in order to make that calculation work you would have to the only way to calculate with certainty would be to have a constant.
ie a way you are going to play at all times.
this does makes the equation quite simple but it then comes up against the loss problem (or put another way, the answer to your initail equation being negative(no profit gained)) and so unless your next bet is adjusted to recoup the loss you will keep making the same bet in CERTAINTY that you WILL NOT profit because of the initial loss and the infinate occurances to come. therefore, the algorithm/method/formula for consistantly profiting must be one that can adjust itself finding a solution to the next question it is posed as a result of the data it has just processed.. In order to do that, an equation must surely have to have contained within it every possible configuration of adjustment bet in relation to the last and future occurances? does that make sense? is it right? given that there are probably many thousands of possible adjusments that could be made given any loss scenario, i view this as an extremely complex concept. Does what i said now make this problem part of complexity theory?

Could you do all the neccesary calculations needed in order to place the next bet in the same time it would take you to work out if the bet you have made is the correct one, given that you cant know your bet is the correct one until you get the next result?

1. Let me make sure we put the P, NP question to bed. All of the questions you are asking have a clear answer. All of the questions you are asking are P (and mathematically fairly simple).

So... let's talk about gambling and odds.

2. In roulette, assuming a fair table, every bet in independent. The math for one bet doesn't change based on what happened on previous bets. A bet is either profitable or not profitable on its own. Any claim that a a series of bets, where either the number one bets on or the proper size of the bets change gives an advantage is mathematically invalid.

I can show you mathematically if you want to propose a system. It is fairly simple to prove with high school algebra.


3, In roulette, the only thing that matters is Expected value of one spin. If you play many times, the amount you win or lose will be the sum of the expected values of each individual spin... the more times you spin, the more likely you will be closer to this number.

4. The math is fairly simple and well known. Any "betting system" that claims to be able to beat mathematical facts is a sucker's game.

If you are going to gamble much, and want to know how much you are likely to win or lose, you need to understand the concept of expected value. Once you know the expected value of one event (i.e. bets rolls or spins)... it is easy to calculate the expected value for repeated events.

If playing a game one time has a negative expected value (and casinos make sure it does because that's how they make money) it is mathematically true that in every case playing multiple times will have a negative expected value.

Again this is fairly easy to show using high school algebra.

This is why I play poker... I am taking money from other players rather than the casino. I only make bets where I am pretty sure I have a positive expected value (which is why I make money playing poker). Of course this only works when other players make bets with a negative expected value... but fortunately there are plenty of players who do.

ok. im going to address your points one by one.
1. Given what we have already said, the following can be established as fact;

#the amount of spins needed to know if you will CONSISTANTLY PROFIT with ABSOLUTE CERTAINTY is INFINITE

#On ANY BET at the roulette table the odds for return are set and expected value are set

#There are 8 DIFFERENT TYPES OF BET YOU CAN MAKE WHICH IF MADE IN COMBINATION WITH EACH OTHER OFFER AN INFINATE MULTIDTUDE OF POSSIBILITIES OF BET FROM FROM SPIN TO SPIN

#REGARDLESS of WHICH BET you make you can never know with ABSOLUTE CERTAINTY which ONE RESULT OUT OF THE 37 OR 38 WILL APPEAR..

#IF a LOSS is made, THEN the ONLY way to RECOUP that LOSS and thereby fulfill the statement clause "CONSISTANTLY PROFIT" is to make a further bet, this is what makes the amount of spins infinate.

# Because the odds are FIXED, IF the return on a CONSTANT BET that you make does not cover any potential loss you may make, then REGARDLESS OF WHETER YOU WIN AT ANY POINT YOU WILL STILL ALWAYS BE IN A NEGATIVE EQUITY POSITION, BECAUSE BY NOT ADJUSTING YOUR NEXT BET TO FACTOR IN THE RECOUPMENT OF YOUR LOSS, YOUR LOSS BECOMES JUST A LOSS AND WILL ALWAYS BE JUST THAT.

#So now, given that in order to profit with ABSOLUTE CERTAINTY, YOU MUST adjust your BET to recoup your losses AND THIS ADUSTMENT MAY HAVE TO BE MADE EVERY TIME YOU MAKE A BET DEPENDING ON WHETHER YOU WIN OR LOSE, AND there are an INFINITE number of ways to do this AND that the configuration for the ORDER in which you are asked make these adjustments is UNKNOWN because it is DETERMINED by a RANDOMNESS which is INCALCUABLE, AND that this must be done an INFINITE AMOUNT OF TIMES.....


WITH SOO MANY VARIABLES AND NO ENDPOINT TO THE CALCULATION, HOW CAN YOU KNOW WITH ABSOLUTE CERTAINTY THAT YOU WILL CONSISTANTLY MAKE A PROFIT?

to me this question has no answer, It is based on a series and set of decision problems where the input to the equation is infinitely variable at each step based upon the last result and also has an infinite runtime..

By the very definition of the word INFINITE, because you would have to play FOREVER, you would never have an ENDPOINT for the equation to either show YES you have consistantly profitted or NO you havent.

THEREFORE, YOU COULD START TO CALCULATE BUT WITH NO ENDPOINT TO YOUR CALCULATION, HOW DO YOU GET AN ANSWER?
AND EVEN IF YOU DID GET AN ANSWER, WOULD YOU BE ABLE TO VERIFY YOUR ANSWER WAS CORRECT IN THE SAME TIME IT TOOK YOU TO FIND THE SOLUTION IN THE FIRST PLACE?

Again, im unsure if what i say is either right or makes true sense, but this question i pose is by no means EASY to answer, i agree that it is P, but given what i put above, i argue that it is in fact a very complex question and subsequent equation that makes it difficult enough to be placed in the NP classification.

I probably sound really dumb, or confused with what i say, and i thank you soo much for bearing with me. Please could you address my points above one by one so that i can properly understand what you are trying to say to me. im a relatively smart guy, but every once in a while need it laid out as if i was a 5 year old in order to fully get what it is im trying to understand..
-------------------------------------------------------
In response to your points 2, 3, and 4, the study of roulette is actually my area of expertise so there is little you can tell me, factually or mythically that i dont already know. It would be unfair of me to say you are wrong and not offer an arguement based on proof, but i disagree with all of what you said but cannot tell you why you are wrong. Please dont take that as an insult or me being ignorant or arrogant, i simply have a different belief to what you and the majority of the people who know the game do.
Id appreciate it if we stuck to the topic of wheter or not the question i posed is related to complexity theory, as your or my opinions on proofs are inconsequential..

Sheesh??
i like that word. By formula i meant an universally applicable method that could factor in all options and respond positivly by itself.
A sort of Artificially Intelligent Equation...
I take it you mean by sheesh, that a formula that could do such a thing would be viewed as incredible...? Almost an Impossibility??

PS. Thanks for the cut and paste. ;p but really wasnt needed..

When I first saw this question, my first thought was "I'll ask him to deposit $5,000 in my bank account and I'll mail him the successful way to avoid losses in gambling". But when I searched for an opportunity to post, everybody was providing very detailed answers. I'm impressed, I wanted to be a smart ass, but the rest of you classed it up so I decided to shut up.

LOL!! Thats funny... Please do join in the conversation, any serious insight or thoughts you may have on the matter will be much appreciated...
Remember though, this is not a conversation about IF there is a way/formula to consistanlty profit, the QUESTION is about HOW COMPLEX that formula would have to be, and whether the wording of the question given the known facts makes the problem of Finding out that formula NP? IF THAT MAKES SENSE..?? ;P