TITLE: How do we know that some radioactive materials have a half life of millions or even billions of years?
QUESTION [20 upvotes]: If a radioactive material takes a very long time to decay, how is its half life measured or calculated? Do we have to actually observe the radioactive material for a very long time to extrapolate its half life?

REPLY [8 votes]: Lubos has already answered your question comprehensively. I would like to add just one thing. It is one of the funny things of the theory of probability that you need not have to wait for that long to confirm the half life. For example it has been estimated that proton has a half life of approximately $10^{32}$ years in some grand unified theories. Now in order to observe proton decay one does not have to wait for $10^{32}$ years. You just take some material which has $10^{32}$ numbers of protons. According to the theory of probability you should observe at least one proton decay per year (if a proton doest indeed take $10^{32}$ years to decay). For an easier example, suppose you throw a dice 6 times. The probability of a particular outcome is same or equivalent to throwing 6 dices at once.
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