# Countably many coin flips probability

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.

• 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. – cr001 Feb 11 at 19:05
• @cr001 Lol, I dare you to do an experiment countably infinite times. – freakish Feb 11 at 19:54