Please check my math on this blackjack sucker bet ("lucky ladies")

Just returned from a casino trip, where I saw a new side bet available in conjunction with the regular blackjack game, called “Lucky Ladies”.

You bet that your first two cards will total 20. The payout varies depending on whether it’s a single-deck or double-deck game, and depending on whether the cards are suited, or matched in number, etc.

The double-deck payout for any 20 was 4:1. As I calculate it, the odds of getting a 20 in double-deck are:

A (10 or face card) as your first card: 32/104, x (10 or face card) as your second card: 31/103.
Plus Ace as your first card, 9 as your second: 8/104 x 8/103
Plus 9 as your first card, Ace as your second: ditto.

Adds up to (992 + 64 + 64)/10712, or .1045. So 10:1 odds against, to win a 4:1 payout.

But what’s REALLY weird, and where I think I must’ve made a mistake, is for the top payout: if you get two Queens of Hearts, it pays 200:1.

That would be (2/104) * (1/103), or 2/10712. 5000:1 odds against? For a 200:1 payout? That is some ridiculous vig.

Did I make a mistake?

Yep. You’re forgetting that the 2xQH also qualifies for the 20-bet.

But remember, the odds always favour the house.

Looks about right. I verified by not considering order (being dealt Qh #1, Qh #2 is equivalent to being dealt Qh #2, Qh #1):

104 Choose 2 = 5356 possible starting hands

32 10-value cards; 32 Choose 2 = 496 Ten-Ten hands
8 Aces, 8 Nines; 64 A9 hands
(496+64)/5356 = 10.5%

The house edge isn’t as ridiculous as you make it sound, though. I presume the way it works is that you place your bet and if you hit one of the lucky hands, you get paid out according to the payout chart. But you aren’t laying down a bet for each type of payout; you’re placing one bet for a chance at every payout. So some significant portion of that 10.5% of the time, you get 4:1 on your money. But some fraction of the time, you’re getting more than this 4:1.

Expected Value = (prob. of handtype1)(payout1) + (prob. of handtype2)(payout2) + … + (.8954)(-1)

I’m guessing that after you consider all the possible payouts, it’s probably still a sucker bet. That’s just my experience with gimmicky casino wagers.

Is the Queen of Hearts payoff part of playing a Lucky Ladies bet, or is it a separate bet one has to make by itself? If the latter, then it’s the most outrageously unfair bet I’ve ever heard of. If the former, though, then it just has the effect of making the Lucky Ladies expectation slightly less unfair (though it’s still most definitely a sucker bet).

The Queen of Hearts payoff is part of the total Lucky Ladies, so one bet covers all possibilities. So as was suggested, the total odds are the aggregate of all the probabilities, so it’s not quite as bad as it seems. But still bad.

Assuming your math is correct, I read this as:

10712 possible outcomes
1118 pay 4:1
2 pay 200:1
9592 lose

Let’s assume a $1 bet on every outcome. That’s $10,712 in action. You win $4,472 in $4 payouts, and $400 in $200 payouts, and you lose $9,592 in $1 losing bets. 4472 + 400 - 9592 = a net result of -$4,720 on $10,712 in action.

Thus, the house advantage on this wager would be a hair over 44%, which (in my understanding) would violate gaming regulations. So somebody’s math is off; either yours or mine.

There were other payoffs in between the any 10 (4:1) and QH/QH (200:1), but I don’t remember all the details. More or less, they were: same suit, 10:1; same denomination, 25:1. So those would need to get added into your calculations, reducing the house advantage somewhat.

I didn’t know there were gaming regulations on the house advantage. You sure about that?