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Now the question is, are you better off betting half of your money so that you can double/split if you win?
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If you want to assume something like a house edge of 1%, you must assume that money is available for doubling/splitting. Otherwise, the house edge is far greater than 1%.
Since people are being silly with numbers, I actually wasted some time on an estimate.
Assume that BJ is a 1:1 game where you win .495 of the time and lose .505 of the time. This makes it a 1% house advantage game, and ignores doubles, splits, and blackjacks.
If you play one hand per month, taking your refund if your one hand loses, your edge is 4.05%. If you're betting $10, you expect to earn $.405. Almost fifty cents! By the way, you'll never earn as much as fifty cents.
If you play two or three hands per month, your edge is 1.550%. Yes, it's the same for even and odd numbers of hands played. For two hands, you expect to make .31005 on $20 wagered, and for three hands, you expect to make .46508 on $30 wagered. .31005/20 and .46508/30 are the same. Fourty-six cents is about the most you can make -- it will go down as more hands are played.
If you play four or five hands, your edge is .925%.
Six or seven, .613%.
Eight or nine, .418%.
Ten or eleven, .281%.
Twelve or thirteen, .179%.
Fourteen or fifteen, .098%.
Sixteen, -.171%.
So if flips negative at sixteen bets. Fifteen bets is a slight winner.
Again, this is an approximation of blackjack, which does not always pay 1:1. The whole .495/.505 thing is an approximation.
I didn't develop a general answer; I just made a spreadsheet layout that let me insert more rows to get the answer. Here's the data for 14 bets:
<font class="small">Code:</font><hr /><pre>$ Real $ Base W L Exp Freq Net Exp EV
140 140 14 0 5.3024E-05 1 5.3024E-05 0.007423365
120 120 13 1 5.40952E-05 14 0.000757333 0.090879985
100 100 12 2 5.51881E-05 91 0.005022114 0.502211364
80 80 11 3 5.6303E-05 364 0.020494282 1.639542554
60 60 10 4 5.74404E-05 1001 0.057497847 3.449870791
40 40 9 5 5.86008E-05 2002 0.117318838 4.692753534
20 20 8 6 5.97847E-05 3003 0.179533374 3.590667477
0 0 7 7 6.09924E-05 3432 0.209326069 0
-18 -20 6 8 6.22246E-05 3003 0.186860519 -3.363489342
-36 -40 5 9 6.34817E-05 2002 0.127090319 -4.575251495
-54 -60 4 10 6.47641E-05 1001 0.0648289 -3.500760614
-72 -80 3 11 6.60725E-05 364 0.024050391 -1.731628146
-90 -100 2 12 6.74073E-05 91 0.006134064 -0.552065791
-108 -120 1 13 6.87691E-05 14 0.000962767 -0.103978825
-126 -140 0 14 7.01583E-05 1 7.01583E-05 -0.008839951
16384 1.00000 0.137
0.098%
</pre><hr />
Explanation:
Each row is one combination of wins/losses; for example, third column is for all outcomes with 12 wins and 2 losses. Assuming a $10 bet, this results in the Base winnings listed. The Real winnings is the same as the Base, but 10% is refunded if the amount is negative.
The Exp shows how often you expect the given combination of wins and losses to hit. For the third row, the number is (.495^12 * .505 ^ 2). (The 12 and 2 are taken from Win and Loss columns.)
Freq shows how many ways there are to get that combination of wins and losses. There's only one way to get 14 wins and no losses (as in row one); if there is one loss, the loss can be the first hand, the second, etc., for 14 ways to have one loss (as in row two). Etc. The number in the column are basically the fourteenth row of Pascal's Triangle. The checksum at the bottom is just the total, which must equal 2^14, which it does.
Net Exp is just Exp * Freq; it shows what fraction of the time that combination hits. The checksum at the bottom is the total, which must equal 1.0, which it does.
EV is the Net Exp frequency times the $Real value; it's the EV in dollars for a $10 bet for that combination. The sum at the bottom -- $.137 -- shows the expected winnings.
Divide the winnings of $.137 by the amount bet of $140 and you get the .098% return.
Mind you, to achieve that return in real blackjack, you would have had to be willing to double and split as appropriate, so the $140 wagered could be more like $180 or $200, reducing the % return, but that's moot -- it's still positive, and it will still flip negative at sixteen bets.
It's really a genius bonus for the casino to offer the players. . . From the casino's perspective, they only give up .01%. A player with a winning month gets nothing, but a player with a losing month gets a nice 10% return, and if feels like 10% to them. For players who haven't tried gambling yet, the idea of 10% back on a loss feels like 10%. Only smart players (who wouldn't play online BJ anyway, except for bonuses) aren't enticed by the phony number. The players who get the biggest "return" are the regular gamblers who are likely to come back and lose month after month, and that's the people the casino really wants to give the most incentive to return.
Genius, I say, genius. If only I could find a similarly effecive way to get donks to migrate to my poker tables. . .