Bonus calculations!

[BradleyT]

I'm coming up with 2 ways to do this and 2 different results so I need some help of a math wiz.

Let's say my theoretical value of a bonus is $100 after meeting wager requirements.

Now let's assume a 1% risk of ruin. In this case I want to use 3,000 hands and 177 wagering units (http://www.wizardofodds.com/games/blackjack/bjapx12.html to verify). We'll say my units are $5. So I need to start with $5 x 177 = $885.

Method 1:
Cashout $985 ninety nine times and lose $885 one time (remember 1% risk of ruin).
$985 x 99 = $97,515 - $885 = $96,630. This was over 100 runs so $96,630 / 100 = $966.30 average cashout for an EV of $81.30 ($966.30 - $885).

Method two:
You'll win $100 ninety nine times and lose $885 one time.
$100 x 99 = $9,900 - $885 = $9,015. This was over 100 runs so $9,015 / 100 = $90.15 average win.

Which method is correct??

[BruceZ]

Method 1:
Cashout $985 ninety nine times and lose $885 one time (remember 1% risk of ruin).
$985 x 99 = $97,515 - $885 = $96,630. This was over 100 runs so $96,630 / 100 = $966.30 average cashout for an EV of $81.30 ($966.30 - $885).

Method two:
You'll win $100 ninety nine times and lose $885 one time.
$100 x 99 = $9,900 - $885 = $9,015. This was over 100 runs so $9,015 / 100 = $90.15 average win.

Which method is correct??


Neither method is correct. In method 1, if you're averaging your cashouts, you don't subtract $885, because when you go broke you cashout for $0, not -$885. Replacing this with 0 gives the same result as the second method, but you are not going to cashout for $985 99 times for a $100 net win. That would be true on average if your EV was exactly 0 before the bonus, but the table you used implies that the house edge before the bonus is 0.414%. If you play 3000 hands at $5/hand, you will bet a total of $15,000, and you will lose on average 0.414% of this or $62.10. Your $100 bonus will bring you up to $37.90. This would be your EV if you always could play to 3000 hands, but that would mean that 1% of the time you would have to find some more money to play with after you lose $885. If you have to quit when you lose $885, then your EV will be reduced since you will not get your bonus. This will happen 1% of the time, so not getting your bonus costs $1 in EV, bringing you to $36.90. Actually, going broke costs less than $1 in EV because when you go broke, you will save the money you would have lost from that point on. You can't go broke before hand 177, so the amount you save when you go broke must be less than 0.414% of $5*(3000-177) or $58.44, and 1% of this is $0.58. So your actual EV is between $36.90 and $37.48. To narrow it down any further would require finding the average number of hands it takes to go broke when you go broke, from which you can calculate the exact amount you save by going broke. We are really after the amount we win assuming we don't go broke, and this is a non-trivial problem.

[BradleyT]

Thanks for correcting my math Bruce!

Now let's see if I comprehend what you did lol.

Let's assume a $250 bonus on a $250 deposit with a wagering require of 30X deposit + bonus ($15,000 again), a house edge of 0.414% again, and again a 1% risk of ruin.

1. Wager requirements X house edge = expected loss
2. Deposit amount X risk or ruin = additional risk
3. Bonus Amount - expected loss - additional risk = Bonus EV

1. $15000 x .00414% = $62.10 (expected loss)
2. $250 x .01 = $2.50 (additional risk)
3. $250 - $62.10 - $2.50 = $185.40 (Bonus EV)

The result is the low end of the EV but in order to keep the calculation super simple is safe enough to use.

Does this look correct? I used my above steps with the figures from my first post and came up with $36.90 so I'm assuming it's pretty close!