To get me started, how do I figure out the probability of quad 7s or better losing when both players use both of their hole cards to make their hand? Assume a 9 max table. I think I could figure it out from here, but here are the payouts as used on Merge.

* 35% to the loser
* 17.5% to the player with the winning hand
* 17.5% to the other players who had cards dealt in the round
* 20% to start the jackpot again
* 10% goes to the poker rooms on the network

There is an extra rake taken (up to an extra $0.50) from each BBJ table. I am trying to figure out if it is +EV to play at a table when the jackpot is $500,000, $1,000,000, etc., actually if I can just find the break even point, that is all that is necessary. Thank you to anyone who helps.

Unfortunately, finding true probabilities is difficult because it's ultimately table dependent. People who fold 33 or 45s preflop will change the overall probability. As a maximum possible likelihood, you can determine the probability of a BBJ hand occurring by assuming all hands go to showdown.

The Wizard of Odds has done this via random simulation for a ten handed table. The probability of quad 7s being beaten (pocket pair needed for quads) is roughly 1 in 164k on any given full table deal. The probability for a 9 handed table will be somewhat less than this. Probably close to 1 in 200k.

http://wizardofodds.com/holdem/badbeat.html

The nasty math behind calculating these odds are illustrated here by Brian Alspach:
http://people.math.sfu.ca/~alspach/comp46.pdf

When you take folds into account, the probability of this will lower somewhat since certain BBJ eligible hands will fold before showdown. I doubt it cuts the probability tons though as most pairs 7s or higher will see a flop and so will a good number of suited connectors.

Now let's consider the combinations of bad beat eligible hands and whether they would usually see a flop: These %ages are pure estimates:

Hand Type: Combos...See flop %age
77...6...90%
88...6...92%
99...6...94%
TT...6...96%
JJ...6...97%
QQ..6...99%
KK...6...100%
AA...6...100%
Total 48 combos

A2s-A5s...16...~45%
23s-26s...16...~20%
34s-37s...16...~23%
45s-48s...16...~26%
56s-59s...16...~29%
67s-6Ts...16...~32%
78s-7Js...16...~37%
89s-8Qs...16...~42%
9Ts-9Ks...16...~52%
TJs-TAs...16...~68%
JQs-JAs...12...~82%
QKs-QAs...8...~96%
AKs...4...100%
Total combos: 184

Total Combos of BBJ eligible hands: 232

Weighted average of BBJ eligible hands seeing the flop: 71.52 combos of SC, 46.08 combos of pairs, overall flop %age: 117.6/232 = ~50.6%


So preflop folds cuts the probability of a BBJ trigger by roughly half at least by raw combination analysis. However, the suited combos that do see a flop will be slightly more likely to trigger the BBJ (e.g. 67s makes more straight flushes than 62s) Post flop folding is even more difficult to determine, but it probably does not happen unless one of the eligible hands needs runner runner to qualify. I would probably assume post flop folding ruins 10% of BBJs at most (and most likely less than this).

Even though I don't have any real math to back this number up, let's see the EV of this bet if the probability of triggering the BBJ if the true probability of triggering it at a 9 handed table is 1 in 450,000 (a reasonable estimate, imo).

Also, another factor is: How many pots actually get large enough to collect the bad beat rake? I'll roughly assume 80% here, if anyone has real data on this, please enlighten us.

With these estimates in mind, this becomes the total table equity where X is the advertised jackpot. The table takes 70% of the advertised jackpot when it's triggered.

(.7)*X/450,000 = $0.5*0.8

X ~ $257,000

This number seems a little low though since Merge's BBJ has got over $1M before.
However, if tables get more shorthanded, BBJ triggering probabilities drop quite quickly. I'm fairly confident that if the BBJ is over $500k though, then it would be worth playing.