The Mathematics of Paradise Poker

I've noticed quite a bit of talk on this board as well as elsewhere about how people have seen way too many "good" hands on Paradise Poker and it is "rigged."

I have always found this hard to believe that it is truly rigged and I provide to you some mathematical evidence to support my belief. If you have evidence which contradicts this, I'd love to see it.

First some basics:

In a 7 card game like those played on Paradise the following probabilities hold true:

A Royal Flush should happen 1 out of every 30,940 hands

Any other Strait Flush should happen 1 out of every 3591 hands

Quads should happen 1 out of every 595 hands


Now, Paradise is one of (if not the) largest online card rooms. From what I've seen I think we can make the following assumptions and have them be on the conservative side.

1. There are on average 200 tables active.

2. There are on average 60/hands/hour on a table.

3. There are 10 people on a table.

4. There are 24 hours in a day.

Therefore, there are on average 2,880,000 hands per day (This is not games, but possible 7 card hands.)


This means in a given day you should see:

93 Royal Flushes, or about 0.5 per table

802 Strait Flushes, or about 4 per table

4840 Quads, or about 24 per table


Now, this assumes that everyone keeps their hands until all 7 cards are dealt, which does not happen, so not all of these hands are seen. At the low limits though where people are apt to keep a pocket pair to the river, or any other odd set of cards, there are quite a few hands seen.


In a live card room you see maybe 25-30/hour if things are going well. At Paradise you see hands going by at least twice as fast, sometimes much faster. It may seem like you are seeing many more "good" hands than you normally would, and when you don't get them it usually ends in some kind of bad beat if you were in the hand.


When playing on-line poker remember, it is an accelerated time frame. You win twice as often and you lose twice as often and you suffer a bad beat to a great hand twice as often.


No one is out to get you or "rig" the game. You just have to get used to the world of on-line poker.

I think you need to check your math, at least where Hold-Em is concerned. With five community cards, at most one Royal Flush, at most two Straight Flushes and at most one set of Quads can be dealt in any given game (unless, of course, all relevant cards are community cards). I don't know that this invalidates your argument, but it does decrease the frequency of hands somewhat.

Valid point.

I was really speaking in generalities and my numbers were more or less off the cuff based on what I've seen on paradise as well as the players/tables stats they have on their web page.

I didn't want to get into the details of the individual games, but you are quite correct.

The stats I present would only hold 100% true for 7 card stud games, in hold 'em you have to deal with the community cards and the effect of each hand being made up of the same 5 cards plus your individual 2. The same hold true with Omaha, but you have 4 individual cards.

I may go back and create more concrete numbers based on actual number of hold 'em, stud, and Omaha tables and games in each type and do the correct odds for each type.


As a whole though, with the numbers rough to begin with I think the general point is still valid. In a 24 hour card room the size of Paradise, with the high rate of hands per hour it is not unreasonable to see the number of "Great" hands that people see.


Thank you for the correction.

Also, keep in mind many players are playing 2 tables at once. Plus the shorthanded games (especially the 5 player max tables) routinely are dealing 120+ hands an hour.


Along with this, random events don't tend to always spread out perfectly. If the average should be, say, getting AA once every 221 hands, you will not get AA every 221 hands, you'll have periods of no AA in 500 hands, then get AA three times in 20 hands, etc...


These would help contribute to the "strange hands" dealt in online poker.


- Tony

Absolutely right, Tony, and this is also why BadBeet's proposal to collect a few thousand boards from Paradise and compare them with the percentages for 5-card hands found in a book is not a valid approach to determining what he would like to determine. BadBeet, if you're reading this, I'm glad you are trying to prove your theory with some real evidence, but you do need to do it correctly, that is, look up an appropriate method for analyzing the randomness of a sample and use that on your data, the percentages for how often something will happen on average do not compare directly to the actual values you will measure, especially for the less likely events. If you flip a coin 1000 times, you may get exactly 500 heads, but you probably won't, nevertheless, if you got 515 heads, you would not be able to conclude that the coin was biased. Likewise, if you roll 2 dice 1000 times, you should get around 28 double-sixes, nonetheless, getting 34 or 35 double-sixes doesn't mean that the dice are biased.

"If you flip a coin 1000 times, you may get exactly 500 heads, but you probably won't, nevertheless, if you got 515 heads, you would not be able to conclude that the coin was biased."


Actually, you "probably will"... but I see your point.


index

No, you really probably won't.


If I flip a coin twice, I have a 50% chance of getting exactly one heads. If I flip the coin 4 times, I can have exactly 2 heads 6 ways, by getting: HHTT,HTHT,HTTH,THHT,THTH,TTHH, but there are 16 possible ways for the 4 flips to turn out, so I have a 6/16 chance of getting exactly 2 heads. This can be written as:


(N/2+1)

SIGMA (i), where N is equal to 4.

i=1


For 6 flips it becomes 10 ways when the first one is heads, 6 ways when the second is heads and the first tails, 3 ways when the first 2 are tails and the third is heads, and 1 way when the first 3 are tails. 10+6+3+1 = 20 ways out of 64 total ways, which is 5/16. This can also be written as:


(N/2+1) i

SIGMA (SIGMA (j)), where N is equal to 6.

i=1 j=1


Or equivalently:


(N/2+2)

SIGMA (iC2), where iC2 means "i choose 2", that is (i(i-1)/2)

i=2


For 8 coin flips, we get another sum:


(N/2+2) i

SIGMA (SIGMA (jC2))

i=2 j=2


which now equals 15+10+6+3+1 + 10+6+3+1 + 6+3+1 + 3+1 + 1 = 70, but there are 256 possible permutations, leaving us with only a 35/128 chance of getting exactly 4 heads.


For 10 flips, we have 7C2+6C2+5C2+4C2+3C2+2C2 + 2*(6C2+5C2+4C2+3C2+2C2) + 3*(5C2+4C2+3C2+2C2) + 4*(4C2+3C2+2C2) + 5*(3C2+2C2) + 6*(2C2) = 56 + 2*35 + 3*20 + 4*10 + 5*4 + 6 = 252, out of 1024 possibilities.


So our odds clearly decrease: 50%, 37.5%, 31.25%, 27.34%, 24.6%, though the rate of decrease slows, I don't see any reason to believe that it curves back up. Hence, it becomes rather unlikely that you will get exactly 500 heads if you flip a coin 1000 times, you will still be more likely to get 500 than you will be to get any other particular number, but you will be much more likely to get a number other than 500 than you will be to get 500.

"Hence, it becomes rather unlikely that you will get exactly 500 heads if you flip a coin 1000 times, you will still be more likely to get 500 than you will be to get any other particular number, but you will be much more likely to get a number other than 500 than you will be to get 500."


That's where I got fouled up... I see now..... good ol' standard deviation...


thanks for the clarification.....


index