Calculating EVs for Steering a Cut-Card

Introduction

In this post i will show how the overall EV from proper steering of a cut-card could be calculated for two select scenarios. This is hardly a comprehensive study but it will be helpful to analyze more complex scenarios.

The following is the list of assumptions used in this analysis:

A) It is assumed that the cutting and steering are perfect, i.e the cut-card always lands where it is aimed at
B) Only Aces and Tens are considered in this study
C) It is assumed that the penalties in expectation values from basic strategy deviation for buffer hands are negligible due to the relatively small bet sizes on those hands.

Case 1: Always Steering Tens and Aces as the first card in the hand

The overall ev in this case would be given by the following equation:

ev_overall=f_ace*ev_ace+f_ten*ev_ten Eq.1

where f_ace and f_ten are respectively the frequencies of spotting an ace and a ten as the last card, and ev_ace and ev_ten are respectively the expectation value of an ace and a ten as the first card in the hand.

For a 6D/DAS/S17 game, ev_ace=50.80% and ev_ten=14.34%
The frequency of spotting an ace (f_ace) as the last card is basically equal to the probability of randomly drawing an ace from a full shoe, so f_ace=24/312=1/13.

Likewise the frequency of spotting a ten (f_ten) would be f_ten=16/52=4/13.

substituting the numbers in the above equation yields an ev_overall=8.32%

Case 2: Steering Tens and Aces as the first card in the hand and as the double card for 9, 10, 11

In this case it is assumed that we only steer a ten or an ace to be the first card if doubling opportunities do not arise.

The overall ev in this case would be given by the following equation

overall_ev=ev_FirstHand+ev_doubling Eq.2

where ev_FirstHand is the expectaion value from steering an ace and a ten to be the first card in the hand, and ev_doubling is from steering an ace and a ten as the double card when the hand has a total of 9, 10, 11.

ev_doubling would have 3 components to it as shown in the following equation:

ev_doubling=ev_doubling_9+ev_doubling_10+ev_doubling_11 Eq.3

For each component we get the following:

ev_doubling_9=f_ace*f_tot_9*ev_20+f_ten*f_tot_9*ev_19 Eq.4


where f_ace and f_ten are the frequencies of spotting an ace and a ten as the last card, f_tot_9 is the frequency of getting a total of 9 on the first two cards, ev_20 and ev_19 are the expectation values of a hand with a total of 20 and 19.

Likewise
ev_doubling_10=f_ace*f_tot_10*ev_21+f_ten*f_tot_10*ev_20 Eq.5

Where f_tot_10 is the frequency of getting a total of 10 on the first two cards and ev_21 is the expectation value of a hand with a total of 21.

For obvious reasons, only the ten is used for doubling an 11 so

ev_doubling_11=f_ten*f_tot_11*ev_21 Eq.6

where f_tot_11 is the frequency of getting a total of 11 on the first two cards.

f_ace=1/13; f_ten_4=4/13; f_tot_9=3.55%; f_tot_10=3.55%; f_tot_11=4.75%; ev_19=26.50%; ev_20=58.10%; ev_21=83.00%

Plugging these numbers into equations 4, 5, 6 yields

ev_doubling_9=0.45%; ev_doubling_10=0.86%; ev_doubling_11=1.21%

Using equation 3, we get ev_doubling=2.52%


The equation for calculating the ev from steering an ace and a ten to be the first card is basically the same as equation one in case 1:

ev_FirstCard=f_ace_c*ev_ace+f_ten_c*ev_10 Eq.7

Where f_ace_c and f_ten_c are the corrected frequencies of seeing an ace and ten. That is because while in case we ALWAYS steer the ace or ten to be the first card in the hand, in this case we only do that if doubling opportunities do not arise. Given that we get the following for corrected frequencies:

f_ace_c=1/13*(1-f_tot_9-f_tot_10)
f_ten_c=4/13*(1-f_tot_9-f_tot_10-f_tot_11)

Using the values of of 2-card hand total frequencies given above we get,

f_ace_c=7.15% and f_ten_c=27.12%, using equation 7 we get

ev_FirstCard=7.52%


Using equation 2 we get : ev_overall=7.52%+2.52%=10.04%

Conclusion

As it was shown for the cases discussed, card steering could theoretically be very profitable. However one needs to keep in mind that practical issues such as the frequency of spotting the last card, cutting accuracy, steering accuracy, and heat could possibly severely reduce the player's advantage.
The study presented above could be extended to cover more card steering cases such as steering different card ranks or steering a ten to the dealer's hand. This would possibly be the subject of future studies.

Oooo....good stuff and very doable at times

That is the conditional expectation, not the overall expectation. That formula will give you the EV for each hand given that the first card is a ten or ace. To get the overall EV you would have to include the EV of receiving every other rank as well as the bet sizes for each card.

Per-hand EV = [EV(Ten) * p(Ten) + EV(Ace) * p(Ace)] * BigBet - EV(9...2) * p(9...2) * SmallBet

So if the player bets $100 on a ten/ace and $10 on all other cards, the win rate would be:

Per-hand EV = (0.5265 * 0.077 + 0.1398 * 0.308) * 100 - (0.0097 + 0.0867 + 0.1769 + 0.1981 + 0.1969 + 0.168 + 0.1453 + 0.1258) * 0.077 * 10

$7.51 on an average bet of $44.66 for an EV of 16.81% per hand. The hourly EV would be lower than that since the opportunity is not available on every hand played. The numbers above are from a SD 3:2 H17 DOA NDAS game so using a flat bet will give an EV of -0.17%. Different games would produce different results. ET Fan made a nice post about this topic a while ago on Schlesinger' board:

http://www.advantageplayer.com/blackjack/forums/bj-main/webbbs.cgi?read=15331 (Archive copy)

-Sonny-

Thanks for the input mod buddy .
I think i need to clarify a couple of things. What i meant by overall EV is the increase in overall ev from using the cut-card technique and perfectly always cutting and steering aces and tens spotted as the last card, to be the first card in the hand.
You are right in the sense that this ev is a conditioned one and it only applies given that the first card is an ace or a ten. But i did specifically mention the condition for case 1 : It is assumed that every time an ace (1/13th of the time) and a ten (4/13th of the time) are seen as the last card in the deck/shoe they are perfectly steered to be the first card in the hand, for this case study only the aces and tens are considered, so in case other card ranks are seen the technique is not applied and no cutting or steering takes place.
One last point please note taht i also make the assumption of neglecting expectation value penalties from sacrificial playing decisions to ensure perfect card steering.


P.S: Dont you wish all discussions were civil just like ours

I wasn't questioning your numbers, I was just trying to expand upon them. Your conclusions are correct based on the situations you describe. I was trying to add a little more detail to those situations in order to show how to apply your numbers to potential playing strategies. I think it's a good example of how quickly the EV from back-steering can be diluted due to imperfect circumstances (as you alluded to). The EV is still high, but the skill and conditions necessary to obtain it are fairly rare. Some people have made good money using this technique, but it is very important to understand how volatile these advanced techniques can be.

-Sonny-

That's what happens when people who respect each other converse. :cool2:

RJT.

Which one of you can ban the other?