For example, that in an infinite amount of coin flips, the event that the result are head k times in a row happens an infinite amount of times.
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Erm, what is your question? (Wild guess, in absence of a clear question, but: relevant, Borel—Cantelli?) – Clement C. May 28 '16 at 15:14
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Yes if you flip an infinite sequence the probability of getting $k$ heads in a row infinitely many times is 1. In fact, most things have probability 1 :) – almagest May 28 '16 at 15:17
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Related: http://math.stackexchange.com/questions/17152/given-an-infinite-number-of-monkeys-and-an-infinite-amount-of-time-would-one-of – Leonidas Lanier May 28 '16 at 15:18
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The first Borel-Cantelli lemma says that if $E_1, E_2,\dots$ is a sequence of events such that $\sum_{i=1}^\infty P(E_i)<\infty$, then the probability that infinitely many $E_i$'s occur is zero. (This is written $P(\limsup_{i \to \infty} E_i)=0$ or $P(E_i \text{ i.o.})$, where "i.o." stands for "infinitely often.")
The second Borel-Cantelli lemma says that if $\sum_{i=1}^\infty P(E_i)=\infty$ and the $E_i$ are independent, then the probability of infinitely many $E_i$'s occuring is one. (That is, $P(\limsup_{i\to\infty}E_i)=1$, or $P(E_i \text{ i.o.})=1$.)
Let $X_i$ be the outcome of the $i$th coin flip. Let
\begin{align*}E_1&=\{X_1=T,X_2=H,\dots, X_{k+1}=H,X_{k+2}=T\}\\ E_2&=\{X_{k+3}=T,X_{k+4}=H, \dots, X_{2(k+2)-1}=H,X_{2(k+2)}=T\}\\ &\vdots\\ E_i&=\{X_{i(k+2)-(k+1)}=T,X_{i(k+2)-k}=H,\dots X_{i(k+2)-1}=H,X_{i(k+2)}=T\}\\ &\vdots \end{align*}
Then the $E_i$ are independent as the coin flips are all independent, and each $E_i$ has probability $\left(\frac{1}{2}\right)^{k+2}$. Since $\sum_{i=1}^\infty \left(\frac{1}{2}\right)^{k+2}=\infty$, with probability $1$ infinitely many $E_i$ occur. Finally, the event that infinitely many $E_i$ occur is contained in the event that infinitely many times there are $k$ heads in a row, so with probability $1$ you will get $k$ heads in a row infinitely many times.
kccu
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1$$\limsup_{i\to\infty}E_i=\bigcap_{i=1}^\infty \bigcup_{j\geq i} E_j$$ So it is the event that infinitely many $E_i$ occur. – kccu May 28 '16 at 16:14