# What is the probability that an event happens an infinite amount of times in infinite trials?

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.

• 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
• 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
• – Lanier Freeman May 28 '16 at 15:18

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.

• How do you define lim sup Ei? – Cristian Desivo May 28 '16 at 16:04
• $$\limsup_{i\to\infty}E_i=\bigcap_{i=1}^\infty \bigcup_{j\geq i} E_j$$ So it is the event that infinitely many $E_i$ occur. – kccu May 28 '16 at 16:14