EV of the following BJ Jackpot promotion

Hi everyone,

in a local casino here in Germany they offer the following Jackpot Promotion for Blackjack from time to time:

At the beginning of a certain month the Jackpot is set to 100€. In order to receive it you have to get dealt a certain 3-card-combination that equals 21, for example, King of diamonds/7 of spades/4 of clubs (Kd7s4c). If the Jackpot is not hit during this first day then another 100€ are added to it up to 3000€ for a 30-day-month. The Jackpot will remain 3000€ until it is hit also past this month. If it is hit while holding for example 2000€ it is reset to 100€.

Now I have two questions:

1. Is it possible to play with an advantage during this promotion with the Jackpot having a certain size? If yes, what size does it need to have if you are betting the minimum flat (5€)?

2. Should you deviate from basic strategy in order to maximize your chances of hitting the jackpot?
Examples with the Kd7s4c mentioned above:
--> split any two face cards with at least one Kd in them?
--> hit hard 17 consisting of Kd7s
--> hit hard 14 against dealer 6 consisting of Kd4c
--> list goes on...

For me it was always fun playing it especially because I did deviate from basic strategy thus splitting 20 which i normally never do thinking that the jackpot justifies it.

Thank you for reading and possibly helping me with this problem!


Rules:
6 Decks
No-Holecard
S17
DAS
Infinite Re-Split
Double on 9-11 only
No Surrender
Split Tens can become Blackjack

Let's assume a single deck. You have a (3/52)*(2/51) chance of getting dealt a jackpot-eligible combination. This simplifies to 6/2652, or one in 442. When you get one such combination, you now have a 1/49 chance of hitting it. So assuming you completely ignore basic strategy, you will hit the jackpot once every (442)*(49) hands, or once every 21,658 hands. So let's say the jackpot is 1000 euro, and you are betting 5 euro. The jackpot is worth an extra 200 bets every 21,658 hands. This is an approximately 0.9% boost to your bottom line. That would be enough to overcome the house edge at just about any blackjack game, even with bad rules.

An accurate answer to your questiion can't be given without knowing what the rules and the number of decks are.

To answer your second question, given that you have a 1/49 chance of hitting the jackpot, you should deviate from basic strategy any time the cost of doing so is less than the gain from going for the jackpot. Obviously, if the jackpot was 50 times your bet or more (in this case, 250 euro), you should ALWAYS hit. If the jackpot was less than 50 times your bet, you should scale back (for instance, if the jackpot was only 100 euro, you probably shouldn't do anything radical like hit hard 17).

Quote: dr3dd

...
1. Is it possible to play with an advantage during this promotion with the Jackpot having a certain size? If yes, what size does it need to have if you are betting the minimum flat (5€)?

2. Should you deviate from basic strategy in order to maximize your chances of hitting the jackpot?
Examples with the Kd7s4c mentioned above:
--> split any two face cards with at least one Kd in them?
--> hit hard 17 consisting of Kd7s
--> hit hard 14 against dealer 6 consisting of Kd4c
--> list goes on... ...

Using the example of Kd7s4c, assuming multiple decks, no-peek rule, S17, double on 9-11 only, and double after split, I find you have an advantage on the second day (betting 5€ with the jackpot at 200€). I find that you should always hit Kd4c. And except on the first day, you should always hit Kd7s. On the first day, I find that you should stand on Kd7s only vs the dealer's two through seven.

Here are my splits (only one card with the special suit) with corresponding minimum jackpot sizes:
K vs 4 2800
K vs 5 2200
K vs 6 1800
7 vs 8 300
7 vs 9 1700
4 vs 2 2000
4 vs 3 1400
4 vs 4 700
4 vs 7 2900

And here are my splits of specially suited pairs:
K vs 2 1900
K vs 3 1700
K vs 4 1400
K vs 5 1100
K vs 6 900
7 vs A 3000
7 vs 8 200
7 vs 9 900
7 vs 10 1600
4 vs 2 1000
4 vs 3 700
4 vs 4 400
4 vs 7 1500
4 vs 8 1600
4 vs 9 1800
4 vs 10 2600

Thank you for this analyses! I appreciate it a lot. Could you tell me how you came up with the numbers? Maybe you can pick out one example and put me thorugh?

I have added the rules of the game in the first post.

Quote: dr3dd

Thank you for this analyses! I appreciate it a lot. Could you tell me how you came up with the numbers? Maybe you can pick out one example and put me thorugh?

I have added the rules of the game in the first post.

Thanks! Also, can split aces become blackjack?

Quote: ChesterDog

Thanks. Also, can split aces become blackjack?

No, they would as usual count as 21.

Quote: dr3dd

...Split Tens can become Blackjack ...

Applying this rule, I would split pairs with the specially suited king at lower jackpots. Here are my latest minimum jackpots for splits with one specially suited king:
K vs 2 2500
K vs 3 2000
K vs 4 1500
K vs 5 900
K vs 6 500
K vs 7 2400

And here are my splits of a pair of specially suited kings:
K vs 2 1300
K vs 3 1000
K vs 4 800
K vs 5 500
K vs 6 300
K vs 7 1200
K vs 8 2100

I used an infinite deck model to do an analysis of the game. An example of the strategy would be standing or hitting hard 17 vs 2. Standing would have an EV of -15%, and hitting with a jackpot of 100 would have an EV of -17%, so I would stand. However, when the prize got to 200, I get an EV of +22%, so I would hit then.

Using the infinite deck model and allowing only one split instead of infinite resplitting and ignoring jackpots on split hands, I find that the game has a positive EV betting 5 euros with a jackpot of at least 200 euros.

To do the splitting strategies, I compared the EV's of standing/hitting vs splitting. For example, standing on ten-ten vs 2 has an EV of 64%, and splitting has an EV of 46%. The probability of getting a jackpot with one king is the probability of getting a special 7 and a special 4, which is about 2*(1/52)*(1/52), so the EV of splitting a Kd-ten would be about 46% + 2*(1/52)*(1/52)*(Jackpot/bet). A jackpot of 1300 would raise that sum to above 64% which would make splitting better than standing then.

Quote: ChesterDog

AThe probability of getting a jackpot with one king is the probability of getting a special 7 and a special 4, which is about (1/52)*(1/52)

I think, it's actually twice that, no?

Quote: weaselman

I think, it's actually twice that, no?

Yes, thanks! I'll edit my post.

Quote: weaselman

I think, it's actually twice that, no?

Thanks to weaselman! Here is my revised splitting strategy for pairs with one special card:
K vs 2 1300
K vs 3 1000
K vs 4 800
K vs 5 500
K vs 6 300
K vs 7 1200
K vs 8 2100

7 vs A 3000
7 vs 8 200
7 vs 9 900
7 vs 10 1600

4 vs 2 1000
4 vs 3 700
4 vs 4 400
4 vs 7 1500
4 vs 8 1600
4 vs 9 1800
4 vs 10 2600

And here are my splits of specially suited pairs:
K vs 2 700
K vs 3 500
K vs 4 400
K vs 5 300
K vs 6 200
K vs 7 600
K vs 8 1100

7 vs A 1500
7 vs 8 100
7 vs 9 500
7 vs 10 800

4 vs 2 500
4 vs 3 400
4 vs 4 200
4 vs 7 800
4 vs 8 800
4 vs 9 900
4 vs 10 1300

prob of hitting jackpot =p
House Edge on the game =h
minimum bet = b

Game is +EV when
JackpotValue > h*b/p

p= 3/52 *2/51*1/50(for SD), rest you should know..

Based on a 10 euro table, the HA only is reduced by a miniscule .0427 (infinite deck) to .0452 (single deck) percent for every 100 euros in the jackpot. Assuming that the house advantage is usually about .50 percent, it becomes a positive EV game at 1,200 Euros. The combination hits every 22,100 - 23,434 hands on average. If the combination always has to add up to 21 and you have two cards in the combination to make the jackpot, I would consider that the odds of hitting that jackpot are 52 to 1. The EV on the jackpot at that point is 1.92 Euros per 100 in the jackpot. You would make a decision I guess adding the EV of the jackpot (which you could get by hitting or doubling down only) to the EV of the decision to calculate the total EV and then compare that to the EV you would get on the correct decision.

Walking through the real life decision, if you are betting 10, and the jackpot is 200, and the hand dealt is K-7, and you need a 4 of spades, and you don't know how many 4 of spades are left in the deck, and the dealer has an 8 up.

The expected value of staying on 10-7 (8 decks, S17) is -.383 x 10 = -3.83
The expected value of hitting a 10-7 is -.502 x 10 = -5.02, but the jackpot EV is 1.92 x 2 = 3.84, making the total EV, -1.18, better than the -3.83 you get on staying.