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Francis Salmon

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Oct 15, 2014, 8:18:07 AM

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Revisiting the classics

Right now, I'm rereading old BJ-books with great pleasure. From time to time I stumble on something that I don't quite agree with.
This was the case on page 59 of "Blackbelt in Blackjack" and I'd like to submit it to this forum.
Snyder writes:" So, to come up with 7 heads out of 10 flips is to be 2 heads away from our expectation of 5. We are well within one standard deviation (3.16)"
This sounds right but if you look at it in an other way, you get a different result:
I assign +1 to tail and -1 to head. That leads to a result of -4 for 7 heads and 3 tails with an expectation of 0. The standard deviation still being 3.16 (sqrt 10), we can see that our result is actually off by MORE than one s.d.
Did the Bishop really go wrong here?

Francis Salmon

Don Schlesinger

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Oct 15, 2014, 11:16:12 AM

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Yes, he's dreadfully wrong

Yes, of course, Snyder is dreadfully wrong. You can't use the formula of s.d. = sqrt(npq)for this type of calculation. Rather the s.d. is one-half of that amount. In this case, it's 1.58 to either side of the mean of 5. The number 10 would represent the WIDTH of one standard deviation from the left side of the mean to the right side of the mean.

But, there is more. Using the normal distribution for small numbers of trials such as 10 isn't very accurate. That's because, if you do 10 million flips of the coin, the probability of obtaining precisely 7 million heads is, for all intents and purposes, zero.

But, the probability of obtaining precisely 7 heads out of 10 is far from negligible. In fact, it's a very significant 11.7%. And, the probability of getting 7 or more heads is 17.2%, which doesn't jibe with the cumulative normal value of 2/1.58 = 1.27 s.d.s, which is "only" 10.2%. That's because a) you have to make an adjustment of subtracting 0.5, to obtain the z-statistic, and, even when you do, b) the answer is not quite the same as the precise calculation.

Don

Don Schlesinger
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Oct 15, 2014, 11:33:01 AM

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You might find this helpful
Great site I've used many, many times:
http://vassarstats.net/binomialX.html
Don

Francis Salmon
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Oct 16, 2014, 3:01:47 AM

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Just what I thought
It also had occurred to me that the Gauss curve wouldn't apply to such a small number of trials.So I calculated the probabilities by checking through all the combinations and I got the 11.7% for exactly 7 heads and 17.2% for 7 or more heads that you mentioned.
I didn't know about the adjustment of 0.5 but anyway, I think in the future I'll use the link you kindly gave.
Thanks for a perfect answer.
Francis Salmon

Don Schlesinger
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Oct 16, 2014, 11:07:11 AM

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Continuity factor
"I didn't know about the adjustment of 0.5 but anyway, I think in the future I'll use the link you kindly gave."
Go here: http://www.statisticshowto.com/what-is-the-continuity-correction-factor/
Excellent explanation.
"Thanks for a perfect answer."
My pleasure.
Don

Gorgon
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Oct 18, 2014, 5:49:48 PM

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dubious acclaim
Snyder was never highly regarded as a Blackjack theoretician and rightfully so.

Dave Jay
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Oct 25, 2014, 2:44:02 PM

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lance humble
He told me to beware of wrist turning dealers with a watch. It could be a sign of cheating. I have no doubt it was relevant for the time. But still, every time I see a dealer looking at their watch it always reminds me, nah she just wants break :)

bjdavid
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Oct 25, 2014, 6:13:53 PM

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The watch thing is probably true
Lance Humble's book had a lot of nonsense in it regarding win rates, BR requirements, etc. However, I think the looking at the watch thing was spot on in regard to pitch games. Don't remember which casino, but it was one of the smaller ones in Reno, around 1984. The minute I sat down I got a double hand. Dealer looked at watch before giving me my card. Had an unusual run of doubles thereafter, dealer looked at watch each time before dealing my card. He never looked at the watch on non-double hands.
I don't remember my results on those hands, but I was getting uneasy and started watching to see an actual peek, but could not. The wrist turn was quick and the arm was moving sideways. On maybe the 5th such hand I asked the dealer what time his watch said. His face and clumsy response indicated to me he was peeking. 15 minutes of that crap was enough.

Dave Jay
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Oct 25, 2014, 8:40:03 PM

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ha
I don't doubt it. I knowingly was cheated once. I lost every hand except for the hands I bet for the dealer. This game has raped me in losses, but then there is another type of loss which gets you thinking "aahh wait a minute here..."
1984. Man, I am humbled (no pun). I would've been carded for sure :).

Gorgon
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Nov 4, 2014, 6:33:36 PM

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true story
I always have them deal with their bare feet. The smell is bad but I don't get cheated.

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