Prop bet vs the "poker is rigged" believers

[-moe-]

This may be the wrong forum, but I couldn't find a better one. Sorry if I misposted.

Anyway, I think I've got an interesting case for a prop bet coming up, and I need some pointers about how to work out the probabilities and questions about statistical significance.

Context: a growing number of people in a regular home game I play in are convinced that online poker is rigged in various ways. I'm getting a bit tired of trying to argue with them, so I thought it would be just smarter to try to make some money off this, through a prop bet. Perhaps some of the smarter guys would be educated through the process aswell (I don't mind if they get better at poker, we're just playing for what amounts to peanuts anyway, and they are my friends.)

Now, the most common complaint lately has been that on Pokerstars there is always an "action twist" on the river card (thereby "Riverstars", obviously). The guys championing this idea, says that they are usually, and way more often than what would be expected by chance and compared to other poker sites, able to guess the river card before it drops.

As an example, if one guy was behind on the flop, then draws out on the turn, the river will way more often than what should be expected cause a re-suckout -- "on Pokerstars only", "a Pokerstars special" &c.

What I'm planning to suggest for the prop bet is therefore the following: shown a number of hand histories from Pokerstars where there is an all-in situation before the river, they will attempt to guess the river card. I take note of their guess(es) -- they may bet a number of river cards if they like, or they can pass that specific hand if they don't find a satisfying way there can be an "action twist".

I then present the list of actual outcomes, and with it go through their guesses and use a poker odds calculator to figure out the accumulated odds of the hits and misses, run this through some statistical analysis and gets some figure indicating what the accumulated chance of getting that amount of correct guesses is.

If they guess better than expectation -- and within a margin of statistical significance -- they win the bet. If not, ie they can not guess significantly better than what you would get by random chance, I win the bet.

What do you guys think, does this sound like a decent idea?

I expect working out the part about accumulated odds will take some work, figuring out how correct they need to be for it to have statistical significance, so I plan to make the bet as expensive as I can while still acceptable to them. Nothing less than USD $1000 (our buy-in at the home game is typically in the $50-$100 range, so we're by no means professional gamblers, so I don't think I'll be able to push it higher than that, even if the "omg rigged!" guys pools together resources).

I'm looking for any input you have on this idea.

If anyone could for instance give me some pointers about how to work out the statistics for finding the accumulated "correctness" of their answer, that would've been much appreciated.

How would I go about to make an estimate about how many hand histories I would have to give them to go through to meet some criteria for statistical significance (within two standard deviations? a P<0.5?)? Again, any input would be much appreciated.

(My plan here is BTW to play a bunch of turbo MTTs on Pokerstars, taking note of the tournament ids so they can double check themselves after concluding the bet that I didn't scam them.

Then go through the HHs, picking all relevant hands, anonymizing screennames so they cannot figure out results of earlier hands from later chip-stacks, snipping away information about what happens at the river, then print out the whole thing and give it to them next time we play at the home game. Then let them go through and suggest cards that they would expect to fall at the river, then go through that and run the calculations.)