Extremely Long Runs in Bernoulli Process

Question

I caught my son counting his ribs during a biology exam. As punishment for this act of cheating, I set him in the corner with a fair coin and told him he must stay in the corner, flipping the coin, until it comes up heads. The expected number of flips to the first success is, of course, 2.

Can I argue that his sentence will be completed in a finite number of coin flips?

Can it be argued that I potentially sentenced him to an eternity of flipping that coin because the probability of an infinitely long run of tails is infinitesimally larger than 0?

Answer 1

I would say that he’s assured to flip heads eventually, but there is no upper bound on the number of flips it will take. The number of flips is finite, however, since the probability that the coin never comes up as tails is zero.

I got a question like this wrong in college analysis. The problem asked us to prove something for the infinite case. I did induction, and my induction was a correct proof that the statement was true for every finite case, all infinitely-many of them, but I did not prove it for the infinite case.

EDIT

My proof would go like this: $$P(\text{eventually heads}) = 1-P(\text{never heads}) $$$$= 1-P(\text{tails forever}) $$$$= 1-\underset{n\rightarrow\infty}{\text{lim}}\overset{n}{\underset{i=1}{\prod}}P(\text{tails}) $$$$= 1-\underset{n\rightarrow\infty}{\text{lim}}\overset{n}{\underset{i=1}{\prod}}\dfrac{1}{2}$$$$=1-\underset{n\rightarrow\infty}{\text{lim}}\dfrac{1}{2^n}$$$$=1-0=1$$

This holds for an unfair coin, unless you give him a coin that only has tails.

Answer 2

"Infinity" is a slippery concept.

You have potentially sentenced your poor son to a very long sentence. Specifically, for every given number $n$ of coin flips, there is a nonzero chance that he will need more than $n$ flips.

Put differently, you can't say that he will have served his sentence with certainty after some number $n$ of flips, even for very large $n$.

The good news is that this chance is infinitesimally small, and drops very quickly with larger $n$. It is the probability of flipping at least $n+1$ tails in a row, which is $\frac{1}{2^{n+1}}$. This drops very quickly.

Your son may take some solace from the fact that there is a better chance he will die from a stray meteorite hit than require more than some (large) number $n_0$ of flips. Then again, he may fail to see the philosophical consolation this affords.

I suggest you commute his sentence after about $n=10$ flips.