All good, but note that I was looking at Hi-Lo running count.
http://www.edwardothorp.com/wp-conte...rEachRound.pdf
[DOES BASIC STRATEGY HAVE THE SAME EXPECTATION FOR EACH ROUND?]
From the paper, seems that it compares the first round with a round starts from any segment of a deck.
It did not consider if there is any tend to high or low cards bias after dealing a round.
This is what I read, from my high school math standard, without fully understand the whole paper.
Originally Posted by peterlee
Gronbog's results do not contradict Mr. Thorp.
We could create a very oversimplified example:
It's certainly far more complicated than this but shows the idea of how round 2 EV = round 1 EV even though more high cards are probable on round 1, outside of some other condition such as a cut card.Code:Assume basic strategy EV = 0.0% Assume deal from top of shoe for round 1 so round 1 EV = 0.0% Assume 2 possibilities for shoe comp & EV using basic strategy for round 2 60% of the time more positive cards played on round 1 causes round 2 EV = -1.0% 40% of the time more negative cards played on round 1 causes round 2 EV = +1.5% Round 2 EV = 0.0%, same as round 1 EV.
k_c
In the part of [Proof],
[Since f is onto and the number of x’s and y’s are the same, namely n!,
then f is one-to-one whence the set of f (x) = y are simply a shuffling or
rearrangement of the x’s.]
[he probability distributions for S are identical in G1 and G2.]
+++
Does it say, not only the expectations are the same, but the cards using for two rounds are equal chance for every rank?
And [the probability distributions for S are identical] is the reason why the expectations for two rounds are the same.
If "high cards used more in the first round" is true , then the second round should has a different chance of having a blackjack, so the probability distributions are no long identical.
If cards are removed randomly and are not subject to the constraints that they necessarily have to comprise a valid round then what Mr. Thorp is saying is right.
You showed the effects of requiring the removed cards to comprise a valid round.
If an exhaustive sim is run using only basic strategy taking care to avoid any cut card effect I'm pretty sure it would validate the computed full shoe basic strategy EV. I'm not that much into simulation but I've checked this to a small extent.
To me this indicates that basic strategy EV for any round = full shoe basic strategy EV absent cut card effect even for the requirement that removed cards comprise a valid round.
k_c
Have corresponded with Thorp recently about this. I agree with k_c and Thorp, who wrote the following:
I believe my conclusions are a corollary of Doob's (nontrivial) more general mathematical result.
To get insight into the issues, consider this simple example:
Randomly shuffled deck of 2 red and two black cards. There are six equally probable sequences.
Draw until first red card appears.
Payoff is +1 for red, -1 for black.
E round 1 is 0.
After first round, six partial decks remain and we know they have exactly one red card.
3 are red card poor, two are neutral and one is red card rich.
Second round: repeat.
Result: E=0 again.
Write it out and see.
Obvious or trivial? OK then prove mathematically the general case for m red cards and n black cards. Doable.
OK then prove the general case for BJ play.
Don again. Notice how we FORCE the first round to end with a red card and yet (perhaps somewhat unintuitively), the EV for the second round, which is red-card poor, is STILL zero!
Don
Posting Permissions
Reply With Quote
Bookmarks