# Accuracy of radioactive dating Sex 24 7 dating

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.

So with just a few atoms, there’s not much you can say with certainty.

However, if you have trillions of trillions of atoms, which is what you’d expect from a sample of Awesomium-1 large enough to see, the law of large numbers kicks in.

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.

If fact, for very large numbers of coins, atoms, whatever, you’ll find that the probability that the system deviates from the average by particular amount becomes vanishingly small.

For example, if you roll one die, there’s an even chance that you’ll roll any number between 1 and 6.

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.

The law of large numbers just makes this intuition a bit more mathematically explicit, and extends it to any kind of random thing that’s repeated many times (one might even be tempted to say a talk about the half-life of an atom.

If you take a **radioactive** atom and wait for it to decay, the half-life is how long you’d have to wait for there to be a 50% chance that it will have decayed.

These measurements follow the same rules; if there’s a 10% chance that a particular atom has decayed, and there are a large number of them, then almost exactly 10% will have decayed.

The law of large numbers works so well, that the main source of error in carbon **dating** comes not from the randomness of the decay of carbon-14, but from the rate at which it is produced.

This isn’t a mysterious force at work; there are just more ways to get a 7 (, , , , , ) than there are to get, say, 3 (, ).

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