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A coin is tossed until, for the first time, the same result appears twice in succession. Define a sample space for this experiment.

The solution in the back of the book is:

$[x_1x_2...x_n: n \ge 1, x_i \in [H, T]; x_i \ne x_{i+1}, 1 \le i \le n-2; x_{n-1}= x_n]$

I don't have a clue how this result was achieved. Besides knowing that the sample space of a fair coin is $[H, T]$ I am completely lost.

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  • $\begingroup$ Think about a sequence of flips that would result in a success here, e.g. HTHTHH or HTT. What do all of these have in common? $\endgroup$ – TSF Feb 21 '19 at 16:19
  • $\begingroup$ $n\geq1$ in it should be changed into $n\geq2$. $\endgroup$ – drhab Feb 21 '19 at 16:28
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It is just a listing of the possible outcomes.

$HH$ and $TT$ if $2$ tosses are needed. This with $P(\{HH\})=P(\{TT\})=\frac14$.

$THH$ and $HTT$ if $3$ tosses are needed. This with $P(\{THH\})=P(\{HTT\})=\frac18$.

$HTHH$ and $THTT$ if $4$ tosses are needed. This with $P(\{HTHH\})=P(\{THTT\})=\frac1{16}$.

Et cetera.

So $\Omega=\{HH,TT,THH,HTT,HTHH,THTT,\cdots\}$ as outcome-space and the $\sigma$-algebra on it is $\wp(\Omega)$.

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Its a list of the first $n$ coin flips given as $x_i$, with the restriction that the only consecutive tosses with the same outcome are $x_n$ and $x_{n-1}$.

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