"Ask the Wizard" mistake?
Quote: Ask the Wizard
... So, the probability of the player having a winning blackjack is 0.0474895 * (1-0.045621) = 0.045323, or once in 22.06 hands. So, your way of playing 22.06 hands at $16.50 each would have an expected loss of 22.06 × $16.50 × .0064=$2.33.
... Next, let’s calculate the expected loss if you bet $25, and wait until the first win to use the coupon. The probability of any win is 42.42%, as found in my blackjack appendix 4. This is not exactly the applicable statistic for this situation, due to complications in splitting, but close enough. So, the expected number of hands to play to have a winning hand is 1/0.4242 = 2.36. The expected loss of betting 2.36 hands of $25 each is 2.36 × $25 × .0064=$0.38, which has a cost 84% less than waiting for a blackjack.
I think, this is wrong. An easy way to see why - if on average you had to lose $0.38 before every win of $25, it would be a game way too good to be real :)
I believe, the correct way to think about it is that the expected loss of 2.36 hands, provided, that neither of them wins, is 2.36*$25=$59, and that is the cost of your waiting for a win.
Now, the expected loss of waiting for blackjack will also be higher (because you will never get a black jack in those hands, so you'll be losing faster then the normal house edge). Ignoring pushes, doubles and splits, you should win about 9.35 (42.42%) of the 22.06 hands, and lose 12.71, so your cost should be (12.71-9.35)*16.50=$55.44, making waiting for a BJ strategy 33% better than cashing in on the first win. (The latter costs you $9 overall, the former - just under $6).
Quote: weaselmanI think, this is wrong. An easy way to see why - if on average you had to lose $0.38 before every win of $25, it would be a game way too good to be real :)
I believe, the correct way to think about it is that the expected loss of 2.36 hands, provided, that neither of them wins, is 2.36*$25=$59, and that is the cost of your waiting for a win.
Now, the expected loss of waiting for blackjack will also be higher (because you will never get a black jack in those hands, so you'll be losing faster then the normal house edge). Ignoring pushes, doubles and splits, you should win about 9.35 (42.42%) of the 22.06 hands, and lose 12.71, so your cost should be (12.71-9.35)*16.50=$55.44, making waiting for a BJ strategy 33% better than cashing in on the first win. (The latter costs you $9 overall, the former - just under $6).
The Wizard's analysis is correct.
He isn't saying that you lose $0.38 and then win $25 from the hand you win: he's saying that if you play until you win a hand, your net expected loss, including the hand that you win, is $0.38. This passes the common sense test. Your numbers do not. If you lose 2.36 units for every win, the average win in blackjack being 1.21 units according to the Wizard's appendix 4, then blackjack has a house edge of 1-2.21/3.36=34.2%.
You make two separate errors.
First, you are expected to not win 1.36 hands before you win a hand. If you were expected to not win 2.36 hands before you win a hand, the probability of winning a hand would be 1/3.36=29.8%, not 1/2.36=42.4%.
Second, if you don't win, that doesn't mean you lose: it means you lose or push. In blackjack you push 8.5% of the time, so if those are all loses, you'd have quite a different house edge. Really, in the case of playing until you win, and ignoring the first mistake, your method is effectively analyzing a blackjack variant where ties lose, blackjack pays even money, there's no doubling or splitting, and the player is using the basic strategy of normal blackjack. This has a house edge of around 15%, which is so far off it's useless.
For something to be well received, I guess it is to be posed as a question and in a manner that could not be seen as a smack to the wizardly face, I think.
Quote: odiousgambit
For something to be well received, I guess it is to be posed as a question and in a manner that could not be seen as a smack to the wizardly face, I think.
I don't know what you are talking about. It is a question.
Quote: weaselmanI don't know what you are talking about. It is a question.
Surely the ? makes it a question!
A ! would be a smack in a face?
Quote: WizardofEngland
Surely the ? makes it a question!
A ! would be a smack in a face?
Yes, I think so. I suspect, this was an attempt at sarcasm, but I am afraid, it is lost on me ...
Quote: WizardofEnglandQuote: weaselmanI don't know what you are talking about. It is a question.
Surely the ? makes it a question!
A ! would be a smack in a face?
This was my attempt at a lighthearted joke, but it looks like that was wasted too ;-)
Or was it?!?
Quote: weaselmanI believe, the correct way to think about it is that the expected loss of 2.36 hands, provided, that neither of them wins, is 2.36*$25=$59, and that is the cost of your waiting for a win.
I stand by my answer on the site. The reason your math is wrong is that you usually lose in blackjack, but the average win is more than one unit, due to splits, doubles, and blackjacks. Also, if you play to the first loss then of the 2.36 hands one will be a win, and 1.36 a loss.
Quote: WizardAlso, if you play to the first loss then of the 2.36 hands one will be a win, and 1.36 a loss.
But doesn't it mean that you'll end up losing the total of 0.36*$25=$9? Why not? Pushes, doubles and blackjacks?
Quote: weaselmanWhy not? Pushes, doubles and blackjacks?
Well, yes.
Quote: ZPPSecond, if you don't win, that doesn't mean you lose: it means you lose or push. In blackjack you push 8.5% of the time, so if those are all losses, you'd have quite a different house edge. Really, in the case of playing until you win, and ignoring the first mistake, your method is effectively analyzing a blackjack variant where ties lose, blackjack pays even money, there's no doubling or splitting, and the player is using the basic strategy of normal blackjack. This has a house edge of around 15%, which is so far off it's useless.
Quote: WizardI stand by my answer on the site.
I stand by my statement that the WoO does not like a "gotcha" tone. The question mark was there, but I think the Wizardly face was still stinging....
Quote: odiousgambitI stand by my statement that the WoO does not like a "gotcha" tone. The question mark was there, but I think the Wizardly face was still stinging....
Doesn't it strike you as a bit condescending to jump in to Wizard's defense like that?
I mean if he did not like my question or its tone, he'd be perfectly capable of voicing his disagreement, and standing up for himself, no?
Quote: weaselmanDoesn't it strike you as a bit condescending to jump in to Wizard's defense like that?
I mean if he did not like my question or its tone, he'd be perfectly capable of voicing his disagreement, and standing up for himself, no?
No offense intended. I consider myself "fired" as the Defender of Wizardly Sensibilities due to controversy while in office [g]
