If an experiment consists in flipping a fair coin $n$ times, then there is a probability of $2^{-n}$ for any particular outcome, say all heads HH...H.

If this experiment is repeated $2^n$ times, each possible outcome has an expected value of times to occur equal to $1.$ And the probability of the all heads outcome, or any other one, will be near $e^{-1} \sim .36$ as it follows a Poisson distribution approximately.

What is the proper way to analyze the experiment of flipping a coin a countably infinite number of times? It seems that if we repeat this experiment a continuum number $c$ of times, each possible outcome should have an expected value of $1$ to occur. Does it remain meaningful to ask for the probability that the all-H outcome occurs? And does this probability remain $e^{-1}$ in that case?

Edit: for possible clarity. Let $t$ be a 'time' from the interval $[0,1].$ Then at time $t$ the experiment is performed to flip a coin $\aleph_0$ times. This provides a natural order to the sequence of experiments.

On further reflection, if the all-H roll is expected to occur $N$ times during the time interval $[0,1],$ then it should be expected to occur $2N$ times during the interval $[0,2].$ Since the experiment is performed an equal number $c$ of times in both cases, it follows that the expected number $N$ of times, if it exists, is either zero or infinite.

  • $\begingroup$ You can't do an experiment uncountably infinite times, because there is a natural order of the experiment and therefore the number of experiments is always countable. And the expected number of occurrence of each possible outcome is infinity in such case. $\endgroup$ – cr001 Feb 11 at 19:05
  • $\begingroup$ @cr001 Lol, I dare you to do an experiment countably infinite times. $\endgroup$ – freakish Feb 11 at 19:54

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