Calculating EV of split hand
<Begin quote>
It took me years to get the splitting pairs correct myself. Cindy of Gambling Tools was very helpful. Peter Griffin also addresses this topic in chapter 11 of the The Theory of Blackjack Let’s say I want to determine the expected value of splitting eights against a dealer 2. Resplitting up to four hands is allowed. Here is how I did it.
1.Take a 2 and two 8’s out of the shoe.
2.Determine the probability that the player will not get a third eight on either hand.
3.Go through all ranks, except 8, subtract that card from the deck, play out a hand with that card and an 8, determine the expected value, and multiply by 2. For each rank determine the probability of that rank, given that the probability of another 8 is zero. Take the dot product of the probability and expected value over each rank.
4.Multiply this dot product by the probability from step 2.
steps 5-13.
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My calculations are not dealing with resplitting at all for simplicity, so point 2 is not necessary, and the next player card can be an 8, no resplitting will happen anyway.
But in step 3, is it absolutely correct just to play out one out two hands with an 8, and then to multiply EV by 2 to account for another split hand with 8?
During an actual game, first split hand is fully played out, and that affects card availability for the second hand and for the dealer.
Does the calculation in step 3 above imply an estimate?
Hope someone can shed some light on this.
Thank you,
DM.
Technically you know more cards when you play the second hand - and this could affect the way you would play the second hand.
I didn't check the numbers, but imagine you split 88 against T in single deck.
First hand receives another 8 - with no resplit available you would hit 88 against T on the first hand. Say in another identical game the first hand would get a 5 and you draw another 5. The second hand would receive the 8 - you would probably want to stand on the second hand since two 5s which would significantly improved your hand had already been revealed and thus are unlikely to be drawn. So in this specific scenario you would play the split hand very different, depending on the other cards you have seen.
To get "absolutely correct" EVs for splits, you would need to traverse the full playing tree of both hands. This basically *squares* your numerical effort for this split EV, and gets worse with evey other resplit. It would be your design decision if the improved EV would be worth the exploding numerical demand.
If you are interested in the topic, you should read
Nairn, John A. "Exact Calculation of Expected Values for Splitting Pairs in Blackjack." http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/splitting.pdf
where he uses an efficient caching algorithm (among other numerical tricks) to calculate split EVs up to 4 hands.
Edit: Note that his paper has been submitted to "Journal of Statistical Computation & Simulation" in 2008, but apparently has never been published yet. Prof. Nairn still lists this paper in his publications list as "submitted", which under good scientific standards would imply it is still under peer-review. The paper might however have been withdrawn or it failed the peer-review process. This does not neccessarily implication of a poor quality (it might simply miss the scope of the journal), but leaves a kind of bad taste. Anyway, don't let it bother you - it's still worthwhile reading.
Quote: MangoJ(it might simply miss the scope of the journal)
This is the most likely reason. I skimmed through it; it looks okay and is probably worthy of publication "somewhere". He also has 100+ publications in his actual academic field.
Quote: zaqHello, I'm trying to calculate EV of splitting only to 2 hands. Have a question about one WizardOfOdds Q&A below.
<Begin quote>
It took me years to get the splitting pairs correct myself. Cindy of Gambling Tools was very helpful. Peter Griffin also addresses this topic in chapter 11 of the The Theory of Blackjack Let’s say I want to determine the expected value of splitting eights against a dealer 2. Resplitting up to four hands is allowed. Here is how I did it.
1.Take a 2 and two 8’s out of the shoe.
2.Determine the probability that the player will not get a third eight on either hand.
3.Go through all ranks, except 8, subtract that card from the deck, play out a hand with that card and an 8, determine the expected value, and multiply by 2. For each rank determine the probability of that rank, given that the probability of another 8 is zero. Take the dot product of the probability and expected value over each rank.
4.Multiply this dot product by the probability from step 2.
steps 5-13.
<End quote>
My calculations are not dealing with resplitting at all for simplicity, so point 2 is not necessary, and the next player card can be an 8, no resplitting will happen anyway.
But in step 3, is it absolutely correct just to play out one out two hands with an 8, and then to multiply EV by 2 to account for another split hand with 8?
During an actual game, first split hand is fully played out, and that affects card availability for the second hand and for the dealer.
Does the calculation in step 3 above imply an estimate?
Hope someone can shed some light on this.
Thank you,
DM.
the spitting is also 'killing' me, i wanted to give it up. but i just cannot.
like 88V2, single deck, split, then 8? , 8? v2
surelly, if the first hand is an 8 added, next hand is 8 less in the shoe.
the possi of first added 8 is 2/49. it is not 1 card. but 1 card*2/49. because we don't know what card it is. then next hand, the 8 is 2-2/48...
by this logic, i got answers totally different from wizard and bjstrat.
i tried many others before. all are far differerent numbers.
but i will keep trying.
thanks for the link of MangoJ, i will read about it soon.
