Standard Deviation of an Ace

we talked before about the advantage of having an Ace as our initial card.

Against unknown dealer cards, an Ace as the player's first card gives an advantage of some 51% (approx!).

How do we calculate the SD associated with that single hand that we are about to play?

If the player bet $50 on this hand, then would be the SD for that hand?

Look forward to any comments & thanks in advance

Matt

hey Matt,
i dunno the answer. thing is say you run a sim for the game in question.
then you know the sd/round sort of thing for a certain way of playing on your part, ie by some count and bet spread and ramp.
i dunno but wouldn't standard deviation per round be what it is regardless of knowing an ace is coming? i mean if you didn't know an ace was coming and one came the standard deviation per round would be what it is.
thing is i guess that would be if you was betting according to some game plan, but if you just all of a sudden change the size of your bet cause you know an ace is coming that seems as if it would be a different matter.
maybe you could look at a sim of the game just flat betting $50 a hand but counting and using say i18. you'd then have a known standard deviation for that sized bet. so then if you know the expectation for when you know the ace is gonna come out and this is evaluation is just for one hand, then wouldn't the standard deviation just be your ev for that hand plus or minus the standard deviation per round?

hey sage, thanks for your comment!
i think i have worked it out, by looking at the questions in this earlier thread..
(Dead link: Std Dev on Multiple Hands) _http://http://www.blackjackinfo.com/bb/showthread.php?t=15014_

i guess the variance of 1.33 of a single blackjack hand does not change regardless what the first card is. Hmm, actually i would imagine that 1.33 is the variance of an average hand - it would seem natural that the variance is higher where the first card is an Ace because a double down is more likely if first hand is an ace, plus there is a relatively high chance of a 3:2 payout.

assuming i was betting $50
then SD would be sqrt($50^2x1.33) = $57.66
and EV would be 0.51x$50 = $25.50

if i was betting 5 hands of $50
then EV would be 0.51x$50 + 4 x(-0.005*$50)
= $24.50
and SD would be
sqrt ($50^2 x (1.33+(5-1)x0.5) x 5)
= $204.02

ok i think this is what i was after.

but could someone comment on whether it is reasonable to use variance of 1.33 if we know that the first card will be an Ace?

I don't have a quick answer for your question, but I do know the answer will not be 1.33.

The variance will be higher with an Ace because of the blackjacks and more frequently doubling opportunities.

Here’s where building your own simulator comes in handy. You could arrange for it to always give the player an ace, or maybe record the results only when the player has an ace. Maybe sagefr0g will be able to answer this question after he finishes his simulator.

Not to put any pressure on you, Mr. Fr0g! :laugh:

hey Canceler ole buddy, ole friend, lol, hey you know i'll be knocking on your cyber door with questions pretty soon don't cha?

so far the program is at over 1,252 lines of code and 13 modules.

still gotta make it complete dealers hands.
gotta get it to draw cards beyond a csm like shuffle routine (ie. it doesn't now draw cards after a hand is complete, it only draws cards after a shuffle like a csm).
gotta get splits to work.
so my guess there will end up being at least forty or more modules and at least four thousand lines of code, there or about once it's done, if it's ever done, lol.

i'm proud of myself though, cause last night i finally broke down and wrote out a flow chart, lol.
so at least i can sort of know what's doing what.:whip:

what ever, i finally found something more frustrating to do than play blackjack, lmao.
trying to teach a machine with less intelligence than a grasshopper how to do something that comes simple for most humans.
that and not being the brightest bulb in the room myself makes for a challenge.

what ever, even if i had it done, Matts problem would be hard to deal with cause like QFIT was saying, heck to get billions of rounds out of a program like this you'd need one heck of a hard drive and it would take forever to run, not to mention apparently the rng capabilities of excel are supposed to suck.

just a sampling of errors i had to deal with in image below...

Caution: I'm too lazy to re-read the article to make sure I'm quoting it correctly :grin: but...

Grosjean's article here:
http://www.beyondcounting.com/pdfs/ACEbjfo1.pdf (Archive copy)

shows the variance with an Ace as the first card in the table on page 14.
It varies based on rules from 1.461 to 1.498.

perfect! thanks very much. i was actually reading the 42.08% article earlier but wasnt able to spot the right lines!
thanks a bunch Ken Smith!

i am glad that the Ace-Variance is not significantly higher than the typical 1.33!!

I'll be glad to help if I can. When you come knocking, you might do it in this thread.



Probably a good idea. I think that's another reason I gave up on mine; I lost "the picture" of how my own program worked, and didn't feel like figuring it out again.

FYI, the papers available on the beyondcounting website are not as up-to-date as the chapters in ExhibitCAA. This does not mean that the articles on the Web site are wrong, but CAA might have some minor additions, editing, or cleaner formatting. On p. 87, exact numbers are provided for different numbers of decks and rules, but basically the variance is 1.5, as stated by K-dog (thanks!).