poli2k01
Joined: Aug 17, 2020
• Posts: 26
December 28th, 2021 at 10:00:49 PM permalink
Hi Fellow gamers/gamblers/mathematicians,

I was wondering how to compute the probability of a CHOP = "chopping Wins and Losses" in Blackjack using optimal playing strategy by Wizard of Odds:

Game rules:
Deck 4-8 decks
Splitting non-Aces cards - Up to 3 splits
Splitting Aces - Up to 2 splits only
Allows Double on any 2 cards
Allows Doubles on splits, Except Aces
BJ pays 1.5 to 1
Player loses only original bet on Dealer BJ
Dealer hits on soft 17
Continuous Shuffler (not diminishing deck)

Example:
A sequence of L, W is considered 1 CHOP
A sequence of W, W, L is not a CHOP
A sequence of W, L, W is considered 1 CHOP

My hunch is something like this can be only computed exactly (at least down to 4 decimal places) through a point in time measure of the EV through the game, so it may not be even possible.

Please enlighten me and thank you for the help
pwcrabb
Joined: May 15, 2010
• Posts: 185
December 28th, 2021 at 11:12:02 PM permalink
Coin toss game with balanced 50/50 probabilities of opposite outcomes is not a perfect model for blackjack with its pushes, splits, doubles, surrenders, player errors, etc. However, coin toss game has the advantage of mathematical tractability. One can easily describe its "chop" behavior of singletons, pairs, triples, quads, et al.

Over infinite trials, the proportion of each cluster size (N) is:

(N) ÷ [2**(N + 1)]

For example, clusters of size 3 of either heads or tails, combined, will comprise 0.1875 of all outcomes.

One may again divide by 2 to describe the respective proportions if heads and tails have equal probabilities.

This simple model is useful for approximating the behaviors of games such a Baccarat and Craps. As real games deviate farther from the model game, its descriptive power degrades.
"I suppose I was mad. Every great genius is mad upon the subject in which he is greatest. The unsuccessful madman is disgraced and called a lunatic." Fitz-James O'Brien, The Diamond Lens (1858)
ChumpChange
Joined: Jun 15, 2018
• Posts: 4284
December 28th, 2021 at 11:25:33 PM permalink
Wasn't there a thread about the probabilities of Baccarat "chop" earlier this month? I think there were calculations to over a dozen in a row. Banker won that calculation, Player came in 2nd, and "chop" came in a close 3rd. As for Blackjack, things are lopsided because there's more losing hands and more double downs, splits, and Blackjacks to make up for it.
charliepatrick
Joined: Jun 17, 2011
• Posts: 2900
December 29th, 2021 at 9:34:58 AM permalink
Quote: poli2k01

...I was wondering how to compute the probability of a CHOP = "chopping Wins and Losses" in Blackjack using optimal playing strategy by Wizard of Odds:...

Without going into the details of the BJ you can probably only get an estimate (since to work out all the possible hand combinations, especially given you need splits, would be almost impossible).

Thus it might even be easier to use infinite decks to get a rough idea or look at the probabilities of all the various hands against dealer up-cards for finite decks and work out how often the dealer makes various points (usually one only needs the EV, so you'd need to do more here).

Now you have to work out how often you get various hands given a split scenario and when their totals create a "Chop" (remember player bust can be one half of this).

None of the above sounds easy, so I suspect it's probably just easier to run simulations and count how many times it happens.