Let the random variable X be in the interval [0, 1/3] if "heads" eventuates on the first coin-toss and in the interval [2/3, 1] if "tails.
Let X be in the lowest third of the aforementioned interval if "heads" on the next toss and in the highest third if "tails".
Let X be in the lowest third of the aforementioned interval if "heads" on the next toss and in the highest third if "tails".
Let X be in the lowest third of the aforementioned interval if "heads" on the next toss and in the highest third if "tails".
et cetera, ad infinitum! Then the probability distribution of X is the Cantor distribution.
It is easy to see by symmetry that the expected value of X is E(X) = 1/2.
The law of total variance can be used to find the variance var(X), as follows. Let Y = 1 or 0 according as "heads" or "tails" appears on the first coin-toss. Then: