iCountNTrack
Well-Known Member
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.
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.
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