Clumpiness of Wins
Dr. Ron you have maybe touched on one of the most important aspect of wagering that is called Deviation.
Once you get to your second bet in any wagering situation your finances fall into one of three categories:
1. YOU HAVE A PROFIT.
2. YOU HAVE A LOSS.
3. YOU ARE EVEN.
The key question is: if I am in a losing or winning streak, does the probability of a reversal of this trend increase or decrease, as my number of bets increases?
Take a look below at this insightfull chart on deviation and coin flipping.
First, when you flip a coin 10 times, you'll get one of 1024
possible combinations of heads and tails. Assuming the coin is
unbiased, each combination is as likely as any other. Your
0110001011 is no more or less apt to occur than 1010101010 or
1111111111. Although yours doesn't seem peculiar in any way, it's
still just one out of 1024 equally probable combinations and
therefore has a chance of 1/1024 -- just like any of the others.
Second, when you flip a coin 10 times, the "expected" number of
heads and tails is five each. But "expected" is a statistical
term, and by no means tells what will actually happen. In fact,
based purely on the ways any given number of heads and tails can
be arranged, you can find the following probabilities:
===========================
Chance of various numbers
of heads in 10 flips
no of ways probability
heads
0 1 0.098%
1 10 0.977%
2 45 4.394%
3 120 11.719%
4 210 20.508%
5 252 24.609%
6 210 20.508%
7 120 11.719%
8 45 4.395%
9 10 0.977%
10 1 0.098%
===========================
The list shows that, in 10 flips, five heads can occur most often
-- 252 out of 1024 ways. This is why five is the "expected"
value. But four or six heads aren't all that unlikely, with 210
ways each. And so on, along the line. There's nothing mystical
behind these figures. Just a matter of counting arrangements.
Zero heads can occur one way: 0000000000. One can occur 10 ways:
1000000000, 0100000000, 0010000000, 0001000000, 0000100000,
0000010000, 0000001000, 0000000100, 0000000010, and 0000000001.
You can try it yourself with other numbers of heads, but the
enumerations get messy so you might prefer to take me at my word.
Of course, what the vast majority of people will fail to see is the whole truth about the fact, that the most probable outcome of ten flips of a coin is 5 heads and 5 tails.
An important aspect of DEVIATION IS THAT WHILE YOU ARE EXPERIENCING A POSITIVE DEVIATION, you can and will have LOSING STREAKS. Of course the same principle applies to what will happen during a NEGATIVE DEVIATION. THE MOST IMPORTANT ISSUE CONFRONTING THE US IS HOW DO YOU COMPENSATE FOR THESE DEVIATIONS, AND IN THE PROCESS MAKE CONSISTENT, PROFITS.
Cheers
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