# Sample Space of a Fair Coin

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.

• 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? – TSF Feb 21 '19 at 16:19
• $n\geq1$ in it should be changed into $n\geq2$. – drhab Feb 21 '19 at 16:28

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 $$\wp(\Omega)$$.

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 $$x_{n-1}$$.