We flip a fair coin infinite number of times independently.

Does the event "For all $n\in \mathbb{N}$, between flip number $3^n$ and flip number $3^{n+1}$ we get $n$ heads in a row" has positive probability?

I thought about using Borel-Cantelli lemma but cannot see how to organize the sequence of events in such a way that I can use the lemma.

  • $\begingroup$ What is the probability for a particular $n$? $\endgroup$ – Henry Feb 7 '18 at 21:31
  • $\begingroup$ @Henry , Well, idk.. a row of n heads can be in the beginning or in the end, so it depends. $\endgroup$ – Eran Feb 7 '18 at 21:43
  • $\begingroup$ What's the probability for one toss? Here probably less than $p=\frac{1}{3}$ for heads will give a zero probability. Argh, I see the coin is fair. $\endgroup$ – dEmigOd Feb 7 '18 at 21:45
  • $\begingroup$ What precisely do you mean by "$n$ heads in a row"? At least one run of at least $n$ heads? Or exactly $n$ heads? Can we also have runs of more than $n$ heads? $\endgroup$ – Robert Israel Feb 7 '18 at 21:51

Define a good event $A_n$ - to be $n$ heads in a row somewhere between toss $3^n$ and $3^{n+1}$.

Define event $E_{n, i}$ to be the event that those $n$ heads in a row happen right after toss $3^{n} + i$, so $i$ could be $0, \ldots, 2\cdot 3^{n}-n$.

Probability of each $E_{n, i}$ is $2^{-n}$.

Let's look at $F_{n, i}$, which will be the event $E_{n, jn}$ did not happen for all $j < i$ and $E_{n, in}$ did happen. $F_{n,i}$ are disjoint. Moreover, $F_{n,i} \subseteq A_n$.

$$\mathbb{P}(F_{n,i}) = (1 - 2^{-n})^{i}\cdot 2^{-n}$$


$$\mathbb{P}\left(\bigcup\limits_{i= 0}^{\frac{2\cdot 3^n - n}{n}} F_{n, i}\right) = 2^{-n} + (1-2^{-n})2^{-n} + \ldots + (1-2^{-n})^{\frac{2\cdot 3^n - n}{n}}2^{-n} $$

For sufficiently large $n$, we can bound it by $\frac{1}{2}\cdot2^{-n}\cdot\frac{1}{1-(1-2^{-n})} = \frac{1}{2}$.

So, $\mathbb{P}(A_n) \geq \frac{1}{2}$, and all of them independent, so you can apply the second lemma.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.