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Using this exact technique (waiting until exactly half of the sample has decayed and then marking that time as the half-life), won’t work for something like Carbon-14, the isotope most famously used for dating things, since Carbon-14 has a half-life of about 5,700 years. The amount of radiation a sample puts out is proportional to the number of particles that haven’t decayed.
So, if a sample is 90% as radioactive as a pure sample, then 10% of it has already decayed.
If you leave for a while, lose track of time, and come back to find that neither atom has decayed, then you can’t say too much about how long it’s been.
less than an hour, but there’s a good chance it’s been more.
However, by creating a “map” of carbon-14 production rates over time we can take these difficulties into account. There’s a quantity called the “variance”, written “σ” or “Var(X)”, that describes how spread out a random variable is. So, for a die, If you have two random variables and you add them together you get a new random variable (same as rolling two dice instead of one). This property is a big part of why variances are used in the first place.
Still, the difficulties aren’t to be found in the randomness of decay which are ironed out very effectively by the law of large numbers. It’s why, for example, large medical studies and surveys are more trusted than small ones. The average also adds, so if the average of one die is 3.5, the average of two together is 7.
The margin for error, as I see it presently, cannot be small.Physicist: The predictability of large numbers of random events is called the “law of large numbers“.It causes the margin of error to be essentially zero when the number of random things becomes very large.: Suppose there is a set of variables whose individual values are probably different, and may be anything larger than zero. If there is a group of radioisotopes whose eventual decay is not predictable on the individual level, I do not understand how a decay constant is measurable.
This question is asked with the intention of understanding basically the decay constant of radiometric dating (although I know the above is not an entirely accurate representation).
If you start with only 2 atoms, then after an hour there’s a 25% chance that both have decayed, a 25% chance that neither have decayed, and a 50% chance that one has decayed.