# How is Cantor set similar to coin tosses?

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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?

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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
$$f:C\rightarrow \{0,2\}^\mathbb{N}: 0.a_1a_2a_3...\rightarrow (a_1,a_2,a_3,...)$$
with inverse
$$f^{-1}:\{0,2\}^\mathbb{N}\rightarrow C:(a_n)_n\rightarrow \sum_{k=1}^{+\infty} a_k 3^{-k}$$

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 $HTHTHTHTHTHT....$.

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