# Discount Gambling

## +EV Field Bet for Two Shoe Card Craps

Posted in +EV, card craps by stephenhow on January 19, 2011

At some California casinos, craps is dealt using two 6-deck shoes, one for each die in a roll. Of course, the rolls in this type of game are not independent of each other, as a simple spreadsheet will show you. What is surprising is that a simple count of the 6-spot cards will yield +EV opportunities in the Field, provided the casino pays triple (3x) on boxcars, and double (2x) on snake-eyes on this bet. I’ve analyzed the Field bet for this game, assuming the house deals 1/2 the shoe before shuffling. The graph below shows the theoretical distribution of Field Bet expectation values (EVs) for this two shoe craps game.

The above graph shows that with exact knowledge of the cards dealt from the shoes, the Field Bet becomes advantageous (+EV) 6.44% of the time. The average advantage of a +EV Field bet using a perfect count (e.g., a spreadsheet) is 1.2%. If the house deals deeper than 50% of the shoe, these results will improve.

I found that a simple running count of the Six-spot cards is fairly good at extracting the edge out of the Field. The running Six-spot count works as follows:

• Count the number of 6-spot cards contained in every three rolls.
• Every third roll, add (1 – three_roll_six_count) to the running count.
• The Field Bet is +EV when the running count is >= 7.

Intuitively, this count make sense, because you expect to see one 6-spot card in three rolls. The running count reflects the “excess” 6-spots in the decks, i.e., how loaded the decks are with 6-spots. A simple spreadsheet shows that the sensitivity of the 6-spot card is huge on the Field, because it pays 3x for boxcars (6,6). The effect of the Aces is very small compared to the 6-spot card.

The relationship between this simple running 6-count and the Field Bet EV was verified by Monte Carlo analysis, as shown in the following results:

The simple count only yields about 60% of the opportunities found by a perfect count (the count is +EV only 3.8% of the time, compared to the theoretical 6.44% limit). So while this is an interesting find, it’d be a little boring to stand around waiting to bet the Field once every 25 rolls or so. In practice, you’d probably wouldn’t even make a bet for 2 out of 3 shoes, but you’d bet often once the shoe went +EV. Overall, the average advantage per Field bet made is a little higher than 1%, and on average you’d make a net profit of 0.058 Field bets per shoe. So, even if you’re betting \$100 on the Field when it goes +EV, you’d only make \$5.80 per shoe. That’s a lot of standing around for sub-minimum wage with a lot of risk. But, if you wanted to “take a shot” at making a big win with a small number of big bets, this might just be the ticket for you (especially when the count gets really good).