stig holmquist: normal fluctuations

[stig holmquist]

In the book "Blackbelt in Blackjack" by A.Snyder it is stated on p.59 that the standard deviation is equal to the square root of the number of trials. Not correct!Rather it is only equal to 1/2 of the stated value in a binomial trial with 1/2 probability of success.
his can be varified experimentally in Feller's book on probability on p.21. He tabulates the actual outcome for 10 trials of 100 coin tosses each . He obtained a range of 39-60 per 10 tosses, with a std. dev. of 15.74 for 1000 tosses or half of the claim in the book by Snyder.

A deviation of 2 from 5 in 10 tosses represents 2/1.58=1.265 std. dev., which covers 79+% of the time, while 2 std. dev. or 2-8 would cover 95%.

The book has conviced me to obey the KISS principle.

Have I misunderstood the book?

[Don Schlesinger]

If you toss a coin 100 times, the mean is 50 heads, and one standard deviation from the mean is the square root of npq, where n = 100, p = 0.5, and q = 0.5.
Thus SD = sqrt25 = 5. Quick way: take half of the square root of 100. Half of 10 is 5.

But stating what the SD of one hand of BJ is has nothing to do with this. It is simply 1.13 times the square root of the number of hands played (assuming flat betting). We aren't considering straying from the mean; rather, we're considering an absolute value.

See BJA, pp. 151-152 for a discussion of why, in this instance, the SD for the 100 coin tosses is sqrt100 = 10, and not 5.

Don

[Cyrus]

"But stating what the SD of one hand of BJ ...."

Looking for trouble again, I see!

[Don Schlesinger]

>"But stating what the SD of one
> hand of BJ ...." Looking for trouble
> again, I see!

Well, what I wrote wasn't terribly elegant, but it's not really wrong. The SD of any number of hands of BJ is 1.13 times the square root of the number of hands in question. If that happens to be one, then it's simply 1.13 times the bet size.

Don

[ET Fan]

... the most glaring error in BB in BJ. Arnold Snyder acknowledged the error in an issue of Blackjack Forum. Haywood made the same error in Beatwebcasinos.com.

The standard deviation is defined as the square root of the average squared deviation from the mean. When you are flipping coins, if you count heads as 1, and tails as 0, the average result for one flip is 0.5. The actual result will be 0 or 1, so the squared deviation will always be .5^2 = .25. SD is the square root of that, or .5.

OTOH, if you are betting even money on the coin flip, the result can be either +1 or -1. The mean, if the coin is fair, is 0, so the SD works out to 1.

Therefore, if you are counting coins, the SD is the square root of the number of flips times .5, but if you are counting your money, it's the square root of the flips times 1 (assuming you're betting $1 each time).

In blackjack, since we counters are GREEDY buzzards, we typically talk about the SD of the original bet. :-}

ETF

[Stig Holmquist]

If I misunderstood the discussion about SD for coin tossing, it must be because he misrepresented
the fundamental laws of probability theory.
No amount of warped arguments or rationalizations can change the fact that the SD for 10 coin tosses must be 1.58. This can most easily be demonstrated or confirmed by counting odd or even digits in five dual digit integers in a random numbers table. To get a reasonably accurate value one must perform at least 30 trials.

[Don Schlesinger]

I just reread p. 59 of Snyder. You are correct. He forgot to divide the square root of the numbers he was studying by 2.

Incredibly, on p. 122 of Chambliss and Roginski's "Fundamentals of BJ," they do exactly the same thing, which I pointed out in the errata sheet for that book.

Don

[Tom George]

[ET Fan]

The 1.13, often quoted, is the standard deviation of the result of a one unit initial bet (not including additional moneys bet for doubles or splits). Ex: If $5 was bet originally, the SD of the result of the hand will be 1.13x$5 = $5.65.

If you allow the bets to vary, the SD will be larger than you might expect (not based on an average bet), since the squared formula is weighted heavily toward the higher bets.

ETF