The EV of the bonus discussed in the previous posts ($100 + $100 / $2500) with a House Advantage of 0.5% is $187.50. The way in which we can calculate this is to assume your bet size is very small - say, 1 cent. This is important because with a bet size that small you will (almost) never bust out, and your EV just becomes your initial bankroll minus the amount the house will take out of all of your bets:
EV = Initial Bankroll - (HA * WR)
In this case, EV = $200 - 0.5% * $2500 = $187.50. QED.
I have previously mentioned that there are ways you can slightly increase your EV by wagering in a certain manner, but only by taking on a ton of risk. For example, take the following strategy:
1) Bet $200 (your entire stack) on the first hand
2) If you win, bet $400 on the second hand
3) If you win, bet $800 on the third hand
4) If you win, bet $1100 (exactly enough to meet your $2500 WR)
I think it is pretty obvious that this strategy is highly variant. What is the EV, though? There are 3 cases:
1) You win all bets and end with $2700. Your EV is the probability this happens times the amount you have at the end. In this case, the probability is (0.4975)^4 = 6.1%. This works out to an EV of $165.40
2) You win the first three bets and lose the fourth. EV = (0.4975)^3 * (0.5025) * $500 = $30.94
3) You lose at some point. EV = $0.
Thus the overall EV of this strategy is $165.40 + $30.94 = $196.34 !!! So, by taking on the risk of going bust 88% of the time, making $500 6% of the time, and making $2700 6% of the time, I've managed to increase my EV by a whopping $8.84 - although in terms of percentage, that is a 10% increase in EV. On the other hand, I'm sure the hourly rate is through the roof since you took (at most) a minute to play those hands. :-)
The reason that EV increases is that most of the time in this strategy you don't have to meet the WR - you've gone bust before the House Advantage can act on all $2,500 of your bets. This seems counterintuitive that the more you go bust the higher your EV, but this is exactly what happens with all betting strategies, not just this one. The EV doesn't increase by much (as shown), but it does indeed increase. This also explains why I had to choose a bet size so small that you would never bust out to truly calculate the initial EV - as soon as you have some non-zero chance of going bust, you may not need to wager the full $2500 and your EV goes up a little bit.
We can use one more example just to prove that I'm not playing tricks: the next bonus I will likely do is a $100 + $125 /$4500 bonus. The strategy of betting very small has an EV of
$225 - 0.5% * $4500 = $202.50.
The strategy of betting $225, $450, $900, $1800, $1125 will satisfy your $4500 WR, with the following EV:
1) Win them all: (0.4975)^5 * $4725 = $144.00
2) Win first 4, lose last one: (0.4975)^4 * (0.5025) * $2475 = $76.19
3) Lose at some point = $0
EV = $144.00 + $76.19 = $220.19
Again, the EV doesn't increase by much in dollar terms, but is almost 17% greater. If you (somehow) have the bankroll to withstand the wild swings this strategy would have, a 17% increase in EV may be significant.
For what it is worth, I have plugged the numbers for this new bonus into my spreadsheet and also used an updated "equivalent hands per hour" number of 375 based on the speed of my last bonus. If I choose to continue to flat-bet $3 it should take me 4 hours with a RoR of 6.4% and a RoL of 22.1%. If I bump it down to $2 the numbers become 6 hours, RoR of 3.1%, RoL of 17.3%.
But, really, I just hope a good poker reload comes along. :-)
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