Here's the reply to Micon that I wrote last night but couldn't post:
1) when, in your opinion, is Bitmillions +EV to buy a bunch of tickets for? i.e. what does the jackpot have to be on each of them to warrant firing? I got a feeling when it first came out that bottom jackpot was very +EV and the middle was too when it was like 200+ coins
2) when, in your opinion, is bitcoinvideopoker +EV to turn autoplay on and just let it grind like a boss? IMO ~ 4500 ish
1) The 3 pots on bitmillions will average around 13288, 51.9 and 0.97 when they pay out (and will pay out half these amounts, since 50% is retained for the next winner). When they are at those levels you can expect to lose the 10% house edge. When they're higher, you can expect to lose less.
The formula to work out how much you can expect to get back from a 0.01 bet is:
(12*pool4 + 4608*pool3 + 328320*pool2 + 137164.80) / 94109400
If we plug in the numbers above, we get:
pool4 = 13287.84; pool3 = 51.905625; pool2 = 0.97133
=> 0.009
ie. your 0.01 bet returns 0.009, as expected.
If we plug in the current pot values:
pool4 = 1093.9234; pool3 = 74.56854; pool2 = 1.741585
=> 0.0113
we see that the game is currently quite (13%) +ev. Every 0.01 you bet you can expect to get 0.0113 back - presumably because both the match-2 and the match-3 pots are three times bigger than their average sizes. The match-2 pot is the most significant pot (37% of your returns come from it, vs. 28% from match-3, 19% from match-4, (and 16% from match-1)) - because they put 20%, 30%, and 40% of all losing bets into the match-4, match-3, and match-2 pots respectively).
The game is always +EV when match-4 is over 66994, when match-3 is over 174.5, or when match-2 is over 2.45, no matter how small the other two pots are:
>>> pool4 = 0; pool3 = 0; pool2 = 2.45; (12*pool4 + 4608*pool3 + 328320*pool2 + 137164.80) / 94109400
0.010004832673463013
>>> pool4 = 0; pool3 = 174.5; pool2 = 0; (12*pool4 + 4608*pool3 + 328320*pool2 + 137164.80) / 94109400
0.010001772405307016
>>> pool4 = 66995; pool3 = 0; pool2 = 0; (12*pool4 + 4608*pool3 + 328320*pool2 + 137164.80) / 94109400
0.01000011476005585
but in general you'll need to put the 3 pot sizes into the above formula. Or use:
http://dooglus.com/bitmillions/to do it for you.
2) As for video poker, it depends how you play. If you play perfectly then it becomes +EV at 4921.1, but the bitcoinvideopoker auto-play plays a basic sub-optimal strategy, so it probably needs to be more like 5100. Here's a hand-wavy justification: Each 1000 in the pot increases the expected return by around 0.495%. Perfect play returns something like 99.55%, so an extra 1000 in the pot (making 5000) is more than enough.
Here's the analysis from wizardofodds.com: