Need help on finding house edge on a game

I'm a floor manager at a casino and we discussed a new game at a meeting. We had some trouble with the maths and I was hoping someone could be of help.

Alright, the game goes like this -

There are six decks on a shuffling machine (so no card is ever discarded)

Dealer draws two cards. One card on blue and on red. The suits matter from highest to lowest - Spade, Heart, Club, and Diamond.

If the dealer draws an equal match which means the same suit and same value, all bets are taken. That's 6 cards in 312.

From the looks of it, there's a 1 in 2 chance of winning when betting either red or blue.

Given the above equal match mechanism, what is the house edge and odds assuming I bet on only red?

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There's an additional mechanic to the game wherein you can bet on the suits. The suit of the card with the highest value will be paid 3 to 1. However, if the values are the same such as 10 of hearts vs. 10 of spades, the bets there will be ignored. Again, if it's an equal match such as 10 of hearts vs. 10 of hearts, all bets are taken.

What is the house edge if I bet on three different suits but leave one empty?

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Thank you!

Welcome to the forum, popopo. I am famously bad at these calculations, so will leave it to the math guys.

However, for clarification (missing these details in your description, which they'll need):

In the first part, the higher of the 2 cards (red or blue) wins, paid 1:1, and the other loses. Yes?

What happens if the board is an unsuited pair? A push, as in the suits below? Or is that where suit ranking kicks in? If so, why the double standard, where it matters for red/blue, but not for suits?

If the red/blue cards are an exact match in suit and rank, all bets (red/blue/suits) lose. Yes?

Suit bets don't require you to choose red/blue; just the suit of the winning card, whichever it is. Yes?

Slightly unusual suit ranking; those I've seen follow bridge rankings of Spade/Heart/Diamond/Club. But not required to follow that.

Suits bets look like the same HE of *1.6077% (5/311 chance of matching from remaining cards) (*=remember I suck at the math; could be right, though).

However, that's only considering the value of the higher card to the player, in isolation. The 2nd card can negate either a win or loss into a push with a rank match, but seems to have an equal effect on either side by proportion. Doesn't seem to affect the HE, as the all-lose opportunities remain the same, just the hit rate of win/lose beyond that.

Each individual suit bet is subject to the HE, so if you bet 3 suits, you're winning a net 1 unit 75% of the time, losing 3 units 25% of the time, essentially a push, considering the winning card in isolation. But on exact matches, you're paying 3x the HE, so over the long run, it seems a slightly losing strategy to bet more than 1 suit.

Once again, any or all of this could be wrong. :)

Unless I'm missing something, house edge is 5/311 = 1.6077% as babs said for either bet. The house only wins when the exact same suit and rank appear.

I also agree that "bridge suit order" should be followed. It's also the suit order used in stud poker games to determine who goes first. So some players will be familiar with this order and may get irked if it's not followed.

From lowest to highest:
Clubs, Diamonds, Hearts, Spades.

Quote: tringlomane

...From lowest to highest:
Clubs, Diamonds, Hearts, Spades.

Trivia time! Anyone know why it's this order?

Quote: Ibeatyouraces

Trivia time! Anyone know why it's this order?

Alphabetical.

Quote: beachbumbabs

Alphabetical.

But in reverse.

Quote: Ibeatyouraces

But in reverse.

Well, yes. What I quoted was lowest to highest, and alphabetical. But I think of them as I listed them earlier, highest to lowest, which is reverse alphabetical. So we're both right. :)

I think this could actually be rather easy, but first a question about the suits bet: Does a shared suit lose even if it is the suit you bet on?

Quote: BBB

What happens if the board is an unsuited pair? A push, as in the suits below? Or is that where suit ranking kicks in? If so, why the double standard, where it matters for red/blue, but not for suits?

I'm going to weigh in on an answer to this, after thinking about it. If the suits ranked for the suits bet, that would make Spades the best bet (as they would always win in an unsuited pair containing them), followed by Hearts, and on down, instead of 4 even chances on suits. So that would justify the double standard (win/lose on red/blue, push for suits bet).

Quote: beachbumbabs

...Spades...would always win in an unsuited pair...

BBB- I think an unsuited pair is a standoff. So it's 50:50 which card is absolutely higher and then 1 in 4 which suit that card is.

However I can see countability issues for the suit bet, which is why (as stated in the pre-amble) it has to be dealt from a CSM.

fwiw I should have a higher card suit pays 2/1, both in the same suit pays 5/1, and both the same pays 15/1. That way the player can win a nice sum if lucky.

Quote: charliepatrick

BBB- I think an unsuited pair is a standoff. So it's 50:50 which card is absolutely higher and then 1 in 4 which suit that card is.

However I can see countability issues for the suit bet, which is why (as stated in the pre-amble) it has to be dealt from a CSM.

fwiw I should have a higher card suit pays 2/1, both in the same suit pays 5/1, and both the same pays 15/1. That way the player can win a nice sum if lucky.

Charlie,

Speaking only for the suited bet, which is a pair standoff, I had asked why they ranked the suits for the red/blue bet, which would make a pair a win/lose (unsuited), not a standoff, but did not apply to the suits bet. I answered my own question by saying that if the suits ranked for the suits bet, it would skew the payoffs, and it would never be correct to bet the diamonds suit strategy-wise, because it would lose to the others when paired.

If they were done the same way (ranked suits to break pair ties), it would be correct to bet all 3 higher suits, as I think that would make that bet +ev, with unsuited pairs occurring 5.7878% of the time (18/311), overcoming the -1.6077%HE of identical pairs, and otherwise a push in the long term. The 5.7878% would be reduced by those pairs that didn't include a diamond loser (and become a standoff), but still wipe out the HE.

Quote:

In the first part, the higher of the 2 cards (red or blue) wins, paid 1:1, and the other loses. Yes?

That's correct. Even money is paid on either red or blue.

Quote:

What happens if the board is an unsuited pair? A push, as in the suits below? Or is that where suit ranking kicks in? If so, why the double standard, where it matters for red/blue, but not for suits?

If the cards are of different suits, say a 9 ♠ vs. 9♦, the suit bets will be pushed while the red/blue side with the 9♠ will be paid even money. This is to make the suit bets fair and not give the spades an advantage. There's a "pair" bet that pays 11 to 1, but that's not part of what we're concerned about.

Quote:

If the red/blue cards are an exact match in suit and rank, all bets (red/blue/suits) lose. Yes?

That's correct. There is, however, a "perfect match" bet that pays 50 to 1. Everything else loses.

Quote:

Suit bets don't require you to choose red/blue; just the suit of the winning card, whichever it is. Yes?

Yes. You may bet on them without having to bet on red/blue.

Quote:

Slightly unusual suit ranking; those I've seen follow bridge rankings of Spade/Heart/Diamond/Club.

It makes the layout look more uniform with red/black/red/black - just a design thing.


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Quote:

Unless I'm missing something, house edge is 5/311 = 1.6077% as babs said for either bet. The house only wins when the exact same suit and rank appear.

The last part is correct. I have a question though, why is it 5/311 and not 6/312?


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Quote:

Does a shared suit lose even if it is the suit you bet on?

No. Again, you may bet on the suit bets even if you don't bet on the red/blue sides.


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Having said all that, I still don't know the house edge should the player bet on 3 suits and leave one empty.