Calculating Sticky Bonuses EV

[tastyflop]

Can you tell me if this is right.

100% up to $100. $4800 total wagering required.

Bet $200 on a blackjack hand. Assume a 50% chance of winning. So 50% of time we lose $100, other 50% we have $400 in account.

Finish rollover of $4600 on bj for an average loss of $23 (4600 *0.005). Leaves us with $377 in account. With $100 bonus removed, we have profited $177.

So, 50% of the time we lose 100. The other 50% we gain $177.

EV = (-100 * .50) + (177 * .50) = $38.5

Is this correct?

Thanks!

[ZeuFreak]

You should add the % of time you hit a blackjack. And the chance to double on BJ is not 50% but 48%(even worse if you bet all your balance and can't double) depending on the rules.

[Halo7]

BJ is a bad example since you win significantly less than 50% of the time due to doubles and splits etc. But if you assume a 50 50 game (like flipping a coin, or the banker bet on Baccarat), then your numbers work. The higher you try to get your balance to, the closer the EV gets to the actual bonus amount. I.e. Taking the coin flip example: If there was no maximum, you could bet 200 on a coin flip, and if you win, then 400 on a coin flip, and if you win, 800 on a coin flip, and if you win, 1600 on a coin flip. If you win that, you will have 3200 that you will then grind off $23 playing BJ. Take out the $100 bonus amount, and you have a profit of $3077. You will only win the 4 coin flips in a row .5*.5*.5*.5=0.0625 or 6.25% of the time. The other 93.75% of the time you will lose $100.

So (0.0625 * 3077) + (93.75 * -100) = $98.56 EV in that situation on the $100 bonus.

I know that is a bit overboard since most wouldn't push it that far and the table maximum would prevent it, but I just wanted to show that as you set your target higher and higher, the EV approaches the bonus amount, even on a sticky.

Halo7

[aka23]

The EV of a sticky bonus can be summarized as:

Average Gain = Bonus * Chance of Bust - Average Wagering * House Edge

Note that house edge is expressed per unit wagered above, not per initial bet.

[Zoozie]

If we make it simple and ignore the loss on the remaining wagering requirement, you will gain 50$ by your play as you correct state. But you can do better by playing a higher risk game with you whole balance given the house edge is still not really high.

In the following I am ignoring house edge and remaining wagering requirement and assuming a 33% chance-of-success-bet will pay 2/1 etc.

In you case you played a 50% game and EV of the bonus was 50% of the bonus.

If you play a 33% game you will have EV of 66% of the bonus.
If you play a 10% game you will have EV of 90% of the bonus.
If you play a 1% game you will have EV of 99% of the bonus.

However going for less than 25% win chance will give a very high variance, so I normally just go for 25%.
The roulette is also very good to determine what range you are going for. You can then wager the remaning bonus requirement on the lowest house egde game you can find which is normally blackjack.
Even though the (european) roulette has a 2.7% house edge is is actually still much to go for a 25% win chance compared to a 50% chance in blackjack.

[tastyflop]

Great info Zoozie. Thanks!

As for playing a 33% game, play on roulette is usually not allowed and will void winnings at may casinos. Is there a game you typically play to make a 33% or 25% game?

[aka23]

[ QUOTE ]
Great info Zoozie. Thanks!

As for playing a 33% game, play on roulette is usually not allowed and will void winnings at may casinos. Is there a game you typically play to make a 33% or 25% game?

[/ QUOTE ]

There are many options. In your original post, you mentioned BJ. If you play BJ with a target of 3x bankroll, your chance of success is slightly below 1/3 = 33%. If you play BJ with a target of 4x bankroll, your chance of success is slightly below 1/4 = 25%. Note that when playing BJ to approach a target gain, you should use moderate to large bets... large enough that you can expect to either bust or reach the target before the EV lost to wagering has a large impact, but small enough that your doubles and splits are not limited.

If you wish to only make a single bet, and you cannot play roulette/craps/sicbo, things get more complicated. There are many poker variations where you have a chance of success in the range you listed. However, these all have a varied payout, which complicates results.