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June 26th, 2017, 05:32 AM   #1
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From: israel

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normal numbers and probability

hello.
Recently, I have started to explorer the normal numbers, and I believe this theorem is true:

let x between 0 and 1 be a fixed normal number in base b. we will write x in base b, and then we will look on the infinite list of statements:

statement 1: the sequence '0' appears in the first b digits of x
statement 2: the sequence '00' appears in the first b^2 digits of x
statement 3: the sequence '000' appears in the first b^3 digits of x
.
.
.
statement n: the sequence '000...0'(n zeros) appears in the first b^n digits of x
.
.
.


my theorem is: for every normal number x and base b, at least one of this statements is true.


I have managed to show that the probability that one of the statement is true for x is 1, but does that mean that I proved my theorem?
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June 26th, 2017, 05:43 AM   #2
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I don't know any NT, so I don't know if this is true or not, but if you've only shown that your conjecture is true for x = 1, then you haven't proved it, since your theorem says every normal number x. However, perhaps you can use that for doing mathematical induction?
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June 26th, 2017, 10:42 AM   #3
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thank you for your answer, but I don't think you understood what I did.
Let me put it in this way: I proved that the probability that I was right is 1.
does it mean that I was right?
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June 26th, 2017, 11:04 AM   #4
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Quote:
Originally Posted by dannyh532 View Post
Let me put it in this way: I proved that the probability that I was right is 1.
does it mean that I was right?
That's a much simpler question. In infinitary probability theory, probability 1 does not mean absolute certainty and probability 0 does not mean absolute impossibility. For example the probability that a randomly selected real number from the unit interval is rational is zero, but there are lots of rational numbers that might be selected.
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June 26th, 2017, 10:03 PM   #5
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So I didn't prove anything? Even if I proved that the probability I was right is 1?
What does it mean for a statement to have probability 1 that it is true?

Last edited by skipjack; June 26th, 2017 at 10:36 PM.
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June 26th, 2017, 10:05 PM   #6
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Originally Posted by dannyh532 View Post
so I didn't prove anything? even if I proved that the probability I was right is 1?
what does it mean for a statement to have probability 1 that it is true?
It means it is almost surely true.
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