Answers using BS for a SD, H17, DA2, NoDAS game
Overkill,
You didn't mention which rule option you wanted for doubling after splitting, so I assumed the more-common NoDAS rule. Also, I assumed the player used two-card composition-dependent B.S., which I obtained from Don Schlesinger's outstanding tome Blackjack Attack, 3rd Edition, Table B3 (page 480). This means that he will DD with 4-4 or 5-3 vs. 6, but not with 6-2 vs. 6.
I ran a 400-million-round CVData sim for a heads-up player getting 5 rounds per shoe. In total, the player played 407,618,008 hands.
1. As shown in the tables below, the player Doubled Down 42,943,756 hands, so the DD rate was 10.535% of the hands, or 10.736% of the rounds.
2. Of the 42,943,756 DD hands, the player won 23,924,897 hands, lost 16,035,082 hands, and tied 2,983,777 hands. Thus, his W/L/T percentages were 55.712%/37.340%/6.948%. If we neglect the ties, he won 59.872% and lost 40.128% of the decisive DD hands.
Hands Hands Hands Hands Units Units Advantage Advantage
Total Won Lost Tied Bet Won/Lost Percent Std. Dev.
Splits 14,790,188 7,232,746 -6,452,491 1,104,951 14,790,190 780,255 5.28% 0.04
Soft DD 8,203,904 4,297,109 -3,448,585 458,210 16,407,810 1,697,048 10.34% 0.05
Hard DD 34,739,852 19,627,788 -12,586,497 2,525,567 69,479,680 13,503,200 19.44% 0.022
All DD 42,943,756 23,924,897 -16,035,082 2,983,777 85,887,490 15,200,240 17.70% 0.02
Percentage of 400,000,000 Rounds
Hands Hands Hands Hands
Total Won Lost Tied
Splits 3.698% 1.808% -1.613% 0.276%
Soft DD 2.051% 1.074% -0.862% 0.115%
Hard DD 8.685% 4.907% -3.147% 0.631%
All DD 10.736% 5.981% -4.009% 0.746%
Percentage of 407,618,008 Hands
Hands Hands Hands Hands
Total Won Lost Tied
Splits 3.628% 1.774% -1.583% 0.271%
Soft DD 2.013% 1.054% -0.846% 0.112%
Hard DD 8.523% 4.815% -3.088% 0.620%
All DD 10.535% 5.869% -3.934% 0.732%
.
3. In my sim, the player received a BJ on 4.829% of the rounds, which means he was a bit "lucky", since the probability of a BJ in SD is 2*(4/52)*(16/51)*100% = 4.826546...%, or once every 20.71875 rounds. Naturally, the BJ probability is higher in SD than in 6D, because of the greater "effect of removal". For example, in 6D the BJ probability is only 2*(24/312)*(96/311)*100% = 4.7489488...%, or once every 21.05729... rounds.
Hope this helps!
Dog Hand