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 June 26th, 2017, 05:32 AM #1 Newbie   Joined: May 2017 From: israel Posts: 4 Thanks: 1 Math Focus: number theory 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?
 June 26th, 2017, 05:43 AM #2 Senior Member   Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics 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?
 June 26th, 2017, 10:42 AM #3 Newbie   Joined: May 2017 From: israel Posts: 4 Thanks: 1 Math Focus: number theory 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?
June 26th, 2017, 11:04 AM   #4
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 Originally Posted by dannyh532 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.

 June 26th, 2017, 10:03 PM #5 Newbie   Joined: May 2017 From: israel Posts: 4 Thanks: 1 Math Focus: number theory 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.
June 26th, 2017, 10:05 PM   #6
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Quote:
 Originally Posted by dannyh532 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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