User:IDI - Revision history

IDI: /* A \pi probability problem */

2013-12-11T09:58:24Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T15:03:24Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T15:03:11Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T15:00:52Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T14:59:29Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T14:54:48Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T14:45:42Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T14:44:22Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T14:43:09Z

‎A \pi probability problem

IDI: /* A \pi probability problem */

2013-11-06T14:41:34Z

‎A \pi probability problem

← Older revision		Revision as of 09:58, 11 December 2013
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−	== A <math>\pi</math> probability problem ==	+	=== Testing ===
		+	[http://www.pretenshn.com/mediawiki/images/9/98/Msv-1.pdf MSV 1]
		+	=== A <math>\pi</math> probability problem ===

	After discussing the possibility of finding any piece of information encoded somewhere in the digits of <math>\pi</math>, Calum, a year 12 student, posed the following problem:		After discussing the possibility of finding any piece of information encoded somewhere in the digits of <math>\pi</math>, Calum, a year 12 student, posed the following problem:

@@ Line 2: / Line 2: @@
 == A <math>\pi</math> probability problem ==
-After discussing the possibility of finding any piece of information encoded somewhere in the digits of <math>\pi</math>, Calum, a year 12 student posed the following problem:
+After discussing the possibility of finding any piece of information encoded somewhere in the digits of <math>\pi</math>, Calum, a year 12 student, posed the following problem:
 "Assuming the digits of <math>\pi</math> form an infinitely long, essentially random sequence with no recurrence, is there definitely at least an initial repeat, in the sense that the first <math>n</math> digits could repeat once before the rest continue 'randomly'?"

@@ Line 2: / Line 2: @@
 == A <math>\pi</math> probability problem ==
-After discussing the possibility of finding any piece of information encoded somewhere in the digits of <math>\pi</math>, a year 12 student posed the following problem:
+After discussing the possibility of finding any piece of information encoded somewhere in the digits of <math>\pi</math>, Calum, a year 12 student posed the following problem:
 "Assuming the digits of <math>\pi</math> form an infinitely long, essentially random sequence with no recurrence, is there definitely at least an initial repeat, in the sense that the first <math>n</math> digits could repeat once before the rest continue 'randomly'?"

← Older revision		Revision as of 15:00, 6 November 2013
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	It turns out <math>\phi(0.1)=0.890010...</math>, so the probability that <math>\pi</math> or any other irrational number ''is'' of the required form is apparently <math>0.109990...</math>. Just under a ninth of irrational numbers have this kind of initial repeat. What we don't know is whether <math>\pi</math> is one of them.		It turns out <math>\phi(0.1)=0.890010...</math>, so the probability that <math>\pi</math> or any other irrational number ''is'' of the required form is apparently <math>0.109990...</math>. Just under a ninth of irrational numbers have this kind of initial repeat. What we don't know is whether <math>\pi</math> is one of them.

−	~~Similarly~~, in any infinite sequence of coin tosses, the probability that the second <math>n</math> results reproduce the first <math>n</math> for at least one <math>n</math> is <math>1-\phi(0.5)=0.71</math>ish.	+	(By similar reasoning, in any infinite sequence of coin tosses, the probability that the second <math>n</math> results reproduce the first <math>n</math> for at least one <math>n</math> is <math>1-\phi(0.5)=0.71</math>ish.)

@@ Line 22: / Line 22: @@
 So, <math>P(2n\text{-digit decimal is not of the required form anywhere})=(1-\frac{1}{10})(1-\frac{1}{10^2})...(1-\frac{1}{10^n})=\Pi_{r=1}^{n}(1-\frac{1}{10^r})</math>.
-For an irrational decimal then, the probability that it is not in the required form is <math>\Pi_{r=1}^{\infty}(1-\frac{1}{10^r})</math>. This is an instance of a named function, called Euler's function, <math>\phi(x)=\Pi_{r=1}^{\infty}(1-x^n)</math>.
+For an irrational decimal then, the probability that it is not in the required form is <math>\prod_{r=1}^{\infty}(1-\frac{1}{10^r})</math>. This is an instance of a named function, called Euler's function, <math>\phi(x)=\prod_{r=1}^{\infty}(1-x^n)</math>.
 It turns out <math>\phi(0.1)=0.890010...</math>, so the probability that <math>\pi</math> or any other irrational number ''is'' of the required form is apparently <math>0.109990...</math>. Just under a ninth of irrational numbers have this kind of initial repeat. What we don't know is whether <math>\pi</math> is one of them.
 Similarly, in any infinite sequence of coin tosses, the probability that the second <math>n</math> results reproduce the first <math>n</math> for at least one <math>n</math> is <math>1-\phi(0.5)=0.71</math>ish.

← Older revision		Revision as of 14:54, 6 November 2013
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	For an irrational decimal then, the probability that it is not in the required form is <math>\Pi_{r=1}^{\infty}(1-\frac{1}{10^r})</math>. This is an instance of a named function, called Euler's function, <math>\phi(x)=\Pi_{r=1}^{\infty}(1-x^n)</math>.		For an irrational decimal then, the probability that it is not in the required form is <math>\Pi_{r=1}^{\infty}(1-\frac{1}{10^r})</math>. This is an instance of a named function, called Euler's function, <math>\phi(x)=\Pi_{r=1}^{\infty}(1-x^n)</math>.

−	It turns out <math>\phi(0.1)=0.~~89001~~</math>, so the probability that <math>\pi</math> or any other irrational number ''is'' of the required form is apparently <math>0.~~10999~~</math> ~~exactly~~. Just under a ninth of irrational numbers ~~then,~~ have this kind of initial repeat. What we don't know is whether <math>\pi</math> is one of them.	+	It turns out <math>\phi(0.1)=0.890010...</math>, so the probability that <math>\pi</math> or any other irrational number ''is'' of the required form is apparently <math>0.109990...</math>. Just under a ninth of irrational numbers have this kind of initial repeat. What we don't know is whether <math>\pi</math> is one of them.
		+
		+	Similarly, in any infinite sequence of coin tosses, the probability that the second <math>n</math> results reproduce the first <math>n</math> for at least one <math>n</math> is <math>1-\phi(0.5)=0.71</math>ish.

← Older revision		Revision as of 14:45, 6 November 2013
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	For an irrational decimal then, the probability that it is not in the required form is <math>\Pi_{r=1}^{\infty}(1-\frac{1}{10^r})</math>. This is an instance of a named function, called Euler's function, <math>\phi(x)=\Pi_{r=1}^{\infty}(1-x^n)</math>.		For an irrational decimal then, the probability that it is not in the required form is <math>\Pi_{r=1}^{\infty}(1-\frac{1}{10^r})</math>. This is an instance of a named function, called Euler's function, <math>\phi(x)=\Pi_{r=1}^{\infty}(1-x^n)</math>.

−	It turns out <math>\phi(0.1)=0.89001</math>, so the probability that <math>\pi</math> or any other irrational number ''is'' of the required form is apparently <math>0.11</math>. Just under a ninth of irrational numbers then, have this kind of initial repeat. What we don't know is whether <math>\pi</math> is one of them.	+	It turns out <math>\phi(0.1)=0.89001</math>, so the probability that <math>\pi</math> or any other irrational number ''is'' of the required form is apparently <math>0.10999</math> exactly. Just under a ninth of irrational numbers then, have this kind of initial repeat. What we don't know is whether <math>\pi</math> is one of them.

@@ Line 4: / Line 4: @@
 After discussing the possibility of finding any piece of information encoded somewhere in the digits of <math>\pi</math>, a year 12 student posed the following problem:
-"Assuming the digits of <math>\pi</math> form an infinitely long, essentially random sequence with no recurrence, is there definitely at least a temporary repeat, in the sense that the ''first'' <math>n</math> digits could repeat ''once'' before the rest continue 'randomly'?"
+"Assuming the digits of <math>\pi</math> form an infinitely long, essentially random sequence with no recurrence, is there definitely at least an initial repeat, in the sense that the first <math>n</math> digits could repeat once before the rest continue 'randomly'?"
 In other words, and generalised, is any irrational certainly in one of the forms <math>aabcdef...</math>, <math>ababcdef...</math>,<math>abcabcdef...</math> and so on?