Last week I had an interesting discussion with a good friend of mine. He had been playing some online poker and suggested that there is a relationship between new subscription/additional money transfer and the cards that you're dealt, i.e. you get good cards to get hooked. The sites would probably be risking a lot if this was true but the problem still fascinates me.
My first approach to this was to ask my friend to define "good cards" and do a simple binomial test. My friend had though a hard time defining what exactly good cards are. If he gets really bad cards he knows to fold while if he gets good cards he knows to go all in – the bad cards are the ones in between.
My other approach would be to calculate the exact probability of each given hand and then to see if it differs from the expected, perhaps using a Wilcoxon signed-rank test since this should detect a different distribution shape as well as a true shift. I guess the hard part is to calculate the exact probability.
The data would consist of the first 0-100 dealt cards compared with 300-400 dealt cards a week later (or a friend that's been on the site for a while).
Question: How would you suggest to approach the issue?
How Texas hold'em works
I'm no expert gamer (I've only played Texas hold'em 3-4 times) but it's fairly simple, you can find more details on the Wikipedia page here.
The main difference from regular poker is that you only get 2 cards at start. You don't get to switch these cards. On the table are another 5 cards face down. By combining your two with the tables 5 you choose the best possible 5 card-poker hand.
For instance if you get 2 aces you have a good start and you will probably go in strong, likewise if you have a 7 and a 2 your chances to win are very slim and you quickly fold. The hard part is perhaps a queen and a 9 where you might end up without anything although your cards are above the "average". You can find a list of the poker hands here.
Best Answer
You can use Sklansky's starting hand ranking to know the strength of a dealt hand from 1 to 8. Generate a random sample with the first 100 hands, another with the next 100 hands and compare them with the Wilcoxon test.