How is Cantor set similar to coin tosses?

  • #1
EnumaElish
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Main Question or Discussion Point

The wikipedia page on the Cantor set states that it is a model for an infinite series of coin tosses. In what sense are recorded coin outcomes similar to a set of points on the real line?
 

Answers and Replies

  • #2
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The Cantor sets can be defined as ##\{H,T\}^\mathbb{N}## equipped with the product topology and with ##\{H,T\}## the discrete space. Instead of using ##H## and ##T##, I might as well write ##\{0,2\}^\mathbb{N}##. An element of the Cantor set is then a sequences ##(x_n)_n## which takes on elements ##0## and ##2##. It is then very easy to find a homeomorphism between this set of sequences and the Cantor set ##C##. Indeed, we set
[tex]f:C\rightarrow \{0,2\}^\mathbb{N}: 0.a_1a_2a_3...\rightarrow (a_1,a_2,a_3,...)[/tex]
with inverse
[tex]f^{-1}:\{0,2\}^\mathbb{N}\rightarrow C:(a_n)_n\rightarrow \sum_{k=1}^{+\infty} a_k 3^{-k}[/tex]

So more practically, given an element of the Cantor set ##C##, we can see this as a sequence of coin tosses by looking at the base-3 representation. The number ##0.0202020202...## is an element of the Cantor set and corresponds to the sequence ##HTHTHTHTHTHT....##.
 
  • #3
EnumaElish
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To push the analogy a little, what would the ternary numbers containing 1's correspond to in the coin toss experiment? Could those be interpreted as "experiments we did not measure" (they are superposed)?
 
  • #4
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They don't really have an interpretation.
 

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