Between the Sheets EV

[cardsharkk04]

In the game Between the Sheets a person will flip up two cards side by side (lets say a Q and a 3) and then you can wager an amount (up to the size of the pot) that the next card will fall in between those two cards. If you win you take that much from the pot, if the next card is higher than a Q or lower than a 3 then you put the amount of your wager in the pot. The only catch is that if the third card is Q or a 3, then you put in double what you wager. How big should the gap be in between the two initial cards in order for a bet to be +EV??

[Siegmund]

Assuming it's a single deck, and the two faced cards are different, you have 6 ways to lose 2 units; 4*G ways to win 1 unit from a gap of size G; and 44-4*G ways to lose 1 unit.

You want EV = 6*(-2)+4G-(44-4G)>0.
-12 + 4G - 44 + 4G > 0
8G - 56 > 0
G > 7.

That is, the game is -EV with a gap of 6 or less, neutral with a gap of 7, +EV with a gap of 8 or more.

Left as exercise to the reader to calculate how often a sufficiently large gap will come up.

[Skipbidder]

Besides the points mentioned by the other poster, there are some other things to consider. What happens if you bet the pot and win? Sometimes people will have a re-ante and keep playing. If you are playing dealer's choice games, then this might just end the game right there. We used to have a re-ante if it was in the first two rounds (or sometimes if it was during the first pass through the deck). If others are not playing optimally, you MIGHT choose to forego a small edge in order to preserve the chance for a larger future edge. We usually played that you were forced to bet at least one unit no matter what. Finally, you should be able to reliably count cards to some degree as well...assuming that you just kept playing through the deck like we did. This could alter a couple of decisions along the way. Unless the stakes you are playing for are large in comparison to your bankroll, you probably should choose to bet pot or nothing on each hand (or 1 unit if forced to bet something). By the way, I used to do this, and my friends HATED it. They stopped calling the game.

[DavidC]

I've heard that this game is played where you can also bet taht the card will be higher than your high card or lower than your low card... any idea how that may alter the game?

Also, I've heard that it's played with everyone's cards face-down, but the rest of the cards face-up (therefore counting really helps).

--Dave.

[ QUOTE ]
Assuming it's a single deck, and the two faced cards are different, you have 6 ways to lose 2 units; 4*G ways to win 1 unit from a gap of size G; and 44-4*G ways to lose 1 unit.

You want EV = 6*(-2)+4G-(44-4G)>0.
-12 + 4G - 44 + 4G > 0
8G - 56 > 0
G > 7.

That is, the game is -EV with a gap of 6 or less, neutral with a gap of 7, +EV with a gap of 8 or more.

Left as exercise to the reader to calculate how often a sufficiently large gap will come up.

[/ QUOTE ]

Where did the 44 come from? Everything else is logical, I just can't seem to put my finger on that number.

[cardsharkk04]

I play without antes, so people can just bet whenever they feel like it. Almost all my friends have the misconception that you should bet a lot on something like K 2, but they think that it is stupid to bet a lot on something like J2 or Q3. Optimal play would be potting everything over a 7 gapper. Which I did last time I played and my friends all laughed at me when when I potted a massive pot with J2 and posted. So they all think I'm an idiot and the game goes on. I'll get em eventually.

The 44 are the number of cards that can come that will win or lose 1 bet.

52 cards in the deck less 2 cards on the board and 6 cards that will lose 2 bets = 44 cards.