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Standard deviation is the square root of the number of trials, divided by 2. It's not difficult to figure out on any pocket calculator that has a square root key. The square root of 100 is 10, so the standard deviation on 100 trials is 10/2 = 5. The square root of 1,000 is approximately 31.6, so the standard deviation on 1,000 trials is 31.6/2, or approximately 15.8.
Once you understand what the square root of a number means to a statistician, you will understand why it is perfectly normal for you to come up with 7 heads on 10 flips of a coin, but nearly impossible for you to come up with 7,000 heads out of 10,000 flips of the same coin.
The standard deviation of 10 = 3.16/2 = 1.58.
The standard deviation of 10,000 = 100/2 = 50.
So, to come up with heads seven times in ten flips, is to be 2 away from the expectation of 5, and well within two standard deviations (3.16).
But to come up with heads on 7,000 of 10,000 flips, is to be 2,000 heads over the expectation of 5,000. And since one standard deviation on 10,000 flips is only 50, this result is 40 standard deviations away from what's expected. Statistically, this is nearly impossible.
Just how impossible is it? Statistically, we expect to be within one standard deviation 68% of the time, within two standard deviations 95% of the time, and within three standard deviations 99.7% of the time. Suffice it to say that if we get a result that is 40 standard deviations away from our expectation, either the coin or the flipper is crooked. You have a much better chance of winning your state lottery than you do of flipping 7,000 heads in 10,000 tries with an honest coin.
All blackjack players must be concerned with normal fluctuations, as they are a crucial factor in the size of the bets you can afford to make. The following guidelines are based upon statistical realities that should be more than enough for most casual players.
In an hour of play, or about one hundred hands, in a dead even game, you generally will not be ahead or behind by more than 20 units. On rare occasions, however, in a single hour of play, you may expect to be ahead or behind as many as 35-40 units.
If you play off and on over a period of a few days—say, ten hours of play, or about a thousand hands—you probably won't be ahead or behind by more than 75 units, but on rare occasions, you might be ahead or behind by 120 units in a one-thousand-hand period. These estimates of fluctuation assume you always bet only one unit on each hand, and that neither you nor the house has any significant long-term advantage.
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So, let's look at some of the practical considerations of bankrolling your play. Essentially, what you are up against is what mathematicians call normal fluctuation.
As shown by our computer simulation results, even when you have a strong advantage over the house, you'll still sometimes lose, because in the short run, anything can happen.
This is true even for the casinos— although the house enjoys a large edge on their slot machines, on any given day some slot players will win more than they lose, which is why people return to the slots. If all slot players lost every time they played, no one would play. All casino games are designed to allow players to go home winners fairly regularly—just not often enough to compensate for their long run losses.
Let's stick with blackjack, though. Assume you learn to play basic strategy, so that you nearly eliminate the house edge. How much can you win or lose due to normal fluctuation?
To answer that, start by imagining that all of your bets are of equal size. Rather than assigning some dollar value, let's say instead that you bet one unit on each hand. We will assume you are in a traditional single-deck Las Vegas Strip game, playing perfect basic strategy, so that for all intents and purposes, the game is dead even. Over the long run, you'd expect to win nothing and lose nothing. It's like flipping a coin.
Of course, if you try flipping a coin a thousand times, and recording the results, you'd be highly unlikely to come up with exactly 500 wins and 500 losses. There are precise mathematical formulas for predicting the limits of normal fluctuation, and with an introductory course in probability and statistics, you would know how to make such estimations. But for now, let's develop some practical guidelines describing the best and worst you might expect due to normal fluctuation.
Statisticians use the term standard deviation to explain variations from the expected result. For instance, if you flip an honest coin 10 times, your expected result is five heads and five tails. If, however, you came up with 7 heads and 3 tails, this would not be indicative that the coin was dishonest. It would be considered a normal fluctuation.
However, if you flipped a coin ten thousand times, and it came up 7,000 heads and only 3,000 tails, it would be very unlikely that this was an honest coin. Even though the ratio of heads to tails has remained 7 to 3, the large number of tosses makes the result highly unlikely.
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