Perhaps you can run a Sim for this. On a recent cruise I didn't enter the casino until the final day at sea for their hybrid 7 hand blackjack tournament when I spotted an interesting twist at a $5 - $200 BJ table. Upon receiving a dealt BJ the winning player pushes a plunger when pushed spins a wheel with 12 pie sections each containing a number 10 thru 300, the total of all 12 sections equals 400. 400 diveded by 12 equals an average 33 per pie section. BJ pays 6/5 but the player receives a $ bonus equal to the section his spin stops. If flat betting $5 The minimum BJ pay would be $6 + $10 bonus or 3/1, the average ($33) bonus + $6 would be an approximate 8/1 pay for BJ. Although unknown I would assume the worst table rules for a Sim: 8 deck CSM, double 10 and 11 only, no DAS, no DOA, split face cards only, no late surrender.
Rookie, if you just want a rough idea (or even an accurate one) of the expected dollar return of playing this game at $5 per hand, you don't need a sim; it can be calculated easily enough. And the inflated BJ bonus is going to so dominate things that the details of the rules won't make any real difference. Roughly - every additional 1:2 in the BJ payoff is worth an additional 2.26% to the player. So compared to a 3:2 game, 8:1 would yield an extra 29.38%. So if you assume the underlying game would have a house edge of 0.5% if it paid 3:2, then by effectively paying 8:1, this jumps to a massive 28.88% player edge! (Call it 28%) That makes a return of $5 * 28% = $1.40 per hand.
I've seen a game similar to this on a cruise that I was on. There are a couple of caveats that I noticed at that time which greatly affect the expectation. The "wheel" was not freely spinning and, as such, was more like a slot machine reel. The final result was usually oh-so-close to the big win, but I never ever saw it hit, despite being in the casino a lot and there always being a commotion at the table when the "wheel" was spun. The actual results were heavily skewed toward the minimum payout. In the game I observed, all of the queens were removed from the 6 decks in the CSM, so you were constantly playing at a true count equivalent of -4. There also seems to be a problem with your description of the "wheel". If it has 12 sections and the top bonus is 300 then, even if the remaining 11 sections were all 10, the total would be 410. If you can clarify these points, it would be helpful. Still, even assuming the worst, I should be able to come up with some numbers. By assuming the worst, I mean to apply the rules you gave with the assumption that the bonus is always 10. I can run it with and without the queens present in case that ends up being the determining factor.
Another minor variation would be whether or not you get to play the plunger game when both player and dealer are dealt a BJ. But, either way, I can't see why a sim is needed. It's easy to figure out the average extra payout per hand, once you know the average extra payout per BJ.
I like to run the sims anyway. It's nice to test it against known results from other sources. Also the removal of the queens will affect the rate of blackjacks.
That's not a problem. The general formula, incorporating the number of tens, aces and cards, is still going to be trivial. In fact, I probably made things seem overly complicated, the way I went about it in my first attempt. In general, we just want either - p(BJ) * <extra payout per BJ> or, if the bonus is only for a winning BJ - p(BJ, and not dealer BJ) * <extra payout per BJ> And that will give us the extra amount to add on to the $5 * <EV of underlying game>. You've then got a formula that's good for any number/composition of decks and any rules (since you can look up or estimate the EV of the underlying game). E.g., the simplest approximation, with no missing queens, would be to assume 1 BJ every 21 hands, and say that the extra EV is $33/21 = $1.57 per hand (assuming the wheel is in fact fair, which your warning suggests may not be a safe assumption!) I'm not trying to knock your love of sims, but whenever there is a simple, generalised formula available, you can apply it in a much more ad hoc fashion, as and when you come across particular variations, without the need to take a laptop with you on every cruise! (Or, for the rest of us, without the need to send Gronbog a request to run a sim!)
I hear what you're saying. The right tool(s) and technique(s) for solving each problem. There a re a lot of things that can be calculated easily and more quickly than setting up a sim. These are the most valuable to me as test cases, since the expected answer is precisely known. On the other hand there is a lot of information out there that is completely bogus and I always take the opportunity to verify/refute almost all data that I run across before using it in a calculation. In this case, I think we need a combination of calculation and simulation. Assuming that Rookie wants to know whether he missed an opportunity or avoided a disaster, we need to know the underlying value of the game and the actual rate of blackjacks for each variant we think might apply. For me, this is best done using my simulator. We can then use your formulas to come up with the value of the bonus under various assumptions to get the final value of each game variant.
I only saw the results of one hand, a winning BJ and made a casual observation as the wheel was spinning without asking questions knowing I wasn't going to play. The total $ amount of all sections was $490 not $400 (typo) and some sections were odd numbers such as 11, 13 etc., the bonus BJ pay was 300 with the remaining sections 10 to 20 with several duplications as I recall. I didn't think of missing 10 value cards but that seems very logical.
OK. I still think that there are some useful results that we can derive from this. We can do this by comparing the best case and worst case conditions. We can then assume that value of the actual game lies somewhere in between. Let's start with the underlying house edge from my simulator (1 billion hands) for the game as presented by rookie789 with and without queens in play: Full 8 decks: House edge: 3.44% No Queens: House edge: 5.37% Now, the probability of a player blackjack for a full 8 decks is (128/416) x (32/415) x 2 = 4.75% (my sim agrees) or 1 in every 21.05 hands. With no queens we get (96/384) x (32/383) x 2 = 4.18% (also confirmed) or 1 in every 23.92 hands. Similarly, the probabilities that the dealer also has blackjack are: Full 8 decks: (127/414) x (31/413) x 2 = 4.61% No Queens: (95/382) x ( 31/381) x 2 = 4.05% So the probabilities of winning player blackjack are: Full 8 decks: 4.75% x (1 - 4.61%) = 4.53% or 1 in every 22.09 hands No Queens: 4.18% x (1 - 4.05%) = 4.01% or 1 in every 24.95 hands Note that 4.53% / 2 = 2.265% which agrees with Colin's figure for the value of a 1:2 bonus on a winning blackjack for 8 full decks. For the no queens game, the value of a 1:2 bonus on a winning blackjack is 4.01 / 2 = 2.005% For the wheel, the worst case assumption is that you always get a bonus of $10. The best case assumption we should make is that the wheel is fair (I highly doubt that the casino would use a wheel that is deliberately better than fair!). In this case the typical spin of the wheel will result in a bonus of $490 / 12 = $40.83. Also note that in order to maximize the benefit in terms of house edge, we would flat bet the minimum of $5. See below how betting more underrmines the advantage of the blackjack bonus to the point where the game is no longer in your favour. With all this information in hand, we can now evaluate the house edge of the various proposed variants of this game. The formula is: (underlying house edge) + (((percentage of winning blackjacks) x (bonus amount)) / original bet)) Code: Underlying Decreased Actual Game Wheel Assumption House Edge Winning BJ% BJ Bonus BJ Bonus Value Original Bet House Edge House Edge ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Full 8 deck Fair 3.44% 4.53% $40.83 $1.85 $5 37% -33.56% (i.e. player edge) Full 8 deck Fair 3.44% 4.53% $40.83 $1.85 $53.78 3.44% 0% (i.e. even game) Full 8 deck Fair 3.44% 4.53% $40.83 $1.85 $200 0.93% 2.51% (i.e. house edge) No Queens Worst 5.37% 4.01% $10.00 ????? $5 ???? ???? I've started the row for the No Queens game with the worst wheel (always $10) and a $5 bet. If you're interested, you should be able to work out the result for whatever variant you want using a spreadsheet. From my completion of that row, even the No Queens game with the worst wheel results in a player advantage for this game of 2.65% (select the empty space with your mouse to see the result).
Yeah, that's essentially the method I had in mind. I just took it for granted that we would be betting the min $5, and it seemed more useful to jump straight to the dollar return from each hand, rather than the percentage. Then you can multiply by the number of hands per hour to see what you could earn. The simulated house edge seems rather high. But then I'm not sure what the 'split face cards only' rule means exactly. I thought perhaps it might mean you can't split unalike tens, which is a rule I have seen. Is there actually a rule where you can't split 1-9? P.S. On my screen, at least, the final, most important column in your table is hidden. I was confused for a while, until I realised I needed to scroll to the right.
I took "split face cards only" to mean that you could only split 10's but that they could be of any rank. The result, for basic strategy, is no splitting. Perhaps I got it wrong. If so, then I hope that rookie789 will correct me. FWIW, I've never seen such a rule in practice.
