What model of probability applies to atomic decay?

It’s not “assigned”, it is measured.

Actually, it’s very unlikely that you’ve “been told that a particular atom” will decay within that time period. Or to put it another way, no one can say with even that much certainty that “a particular atom” will decay.

What we do know, as @Rarebear says, “through measurement”, is that a given mass of the isotope will decay at a particular rate so that within 5730 years (I’m accepting your figures without attempting to verify them) half of that mass will have “decayed”, meaning that the particular mass of that isotope has one-half of the radioactivity that it had at the start of the measuring period (today). But no one can make predictions on an atomic level.

Thanks, CW. That was my understanding of what half-life meant, but the person who told me this really did extend it to individual atoms. I guess he inferred that since half the mass would decay within that time, that had to mean that any given individual atom within that mass had a chance of decaying within that time. I guess we could apply the frequency model after all if we look at the entire collection of atoms. We could say that the chance of a particular atom decaying is analogous to a particular card being drawn randomly from a deck (if we were to draw 26 cards from a 52-card deck over a period of 5730 years).

@CWOTUS Is correct. You can’t tell with a particular atom. Half life refers to a big population of atoms.

The model is based on the decay rate being proportional to the size of the sample. It is called an exponential distribution, because if you looked at number of particles decaying as a function of time, you would see an exponential curve. If you are familiar with calculus, you get an exponential curve when the f’(x) = k*f(x) for some constant k.

We use global distribution models all the time to estimate things like traffic accidents and population growth. You can’t say for certain who will be involved in an auto accident or who will be giving birth, but you can make a pretty good estimate of the overall numbers.

I think these are all statements of the same thing. A half-life of X years means that each individual atom has a 50% probability of decaying within X years. Each atom could decay in the next 5 minutes or remain stable for the next billion years—obviously you can’t predict a probabilistic event. In a large sample of atoms, however, you can precisely measure the probability of a decay event, characterized by half-life. It is assumed that the same probability of decay applies to every like atom – otherwise none of it makes sense.

Quantum mechanics is highly successful at predicting probabilities; i.e., good agreement (to many decimal places) between theory and experiment.

@submariner: It seems that the frequency model of probability, the one that we use to assess the odds in games of chance, cannot meaningfully be applied here. I’m not sure I understand the question. There is one mathematical theory of probability that applies to both quantum mechanics and blackjack, though the calculations are quite different.

I asked a similar question two years ago and got some thought provoking answers.

Premise: 50% of the whole mass will decay within a given time.
Conclusion: Any given atom within that mass has a 50% chance of decaying within that time.

Is this a valid inference, or does it commit the fallacy of division (i.e., it incorrectly assumes that the part has the properties of the whole)? Now I’m not sure. Lostinparadise’s answer makes me lean toward saying we cannot accept the above inference, strictly speaking.

gasman: probability is not just a mathematical question. It also involves semantics, metaphysics, and epistemology. There have been attempts to make sense of probability and randomness in other ways besides the frequency model.

Thanks, flutherother. I’ll have to check out some of the links in that previous thread.

Well, if half the mass decays in 5730 years, then the other half is left. Some of those atoms might last another 5730 years (or more). Kind of freaky that some C-14 atoms decay right away and some last >10,000 years?

Empirical observation. No one has found a universal formula yet where you enter the number of protons and neutrons and it spits out the half life.

One important observation is the fact that all atoms of any particular isotope are the same, no matter how long they have gone without decaying. An atom that has lived for 10,000 years has the same chance of decay as one that has existed for 1 year. Because of this, atom decay follows a half-life formula. After X years, on average ½ of the atoms have decayed. Since the ½ of atoms that have not decayed still have the same decay rate, it means that after X more years, on average only half of those will decay.

If X is the half life, then the average lifetime of an atom is 2X and the median lifetime is of course X.

What puzzles me is why the individual atoms of an isotope should behave differently when they are identical. If all the atoms are identical they should decay at precisely the same time. And how do atoms know what their half life is?

@flutherother That’s the beauty of quantum mechnics.

The average lifetime is 2x. Is there a known maximum, either in principle or observed so far? What about those atoms with short half-lives—have any been observed to last much longer than 2x?

@submariner No matter the length of the half life, ¼ of atoms life past 2X. 1/8 live past 3X. 1/1024 atoms will live past 10X. If you have one mole of a substance, it is possible that one of the atoms will live past 78X.