Half Life Probability
Date: Spring 2012
I have a question about radioactivity. I have come to understand that the event of decay after one half life is up is a completely random event, and it is also an independent event. On a graph of probability of decay vs time, there will certainly be a point at (1 half life, 50%), and a point at (2 half lives, 50%), assuming that the atom did not decay at the first half life, and so on. It is important to note that this graph is for a single atom, and shows the probability of the INDIVIDUAL events at each half life, and not the probability of the atom "surviving" through several half lives (which of course would simply be a curve). What I want to know is that is it possible to graph any data between any of the points? In other words does the probability vary at some points, then returning to 50% at the next half life?
Even though it may appear that the graph you see is the decay time for an individual atom, it is actually for the “mean” decay time. “Mean” in mathematics (and Physics) implies an average over a collection. If we could trap a radioactive atom and wait for it to decay, the “half life” would have little predictive power in saying when it will decay. The atom may have a half life of 1 second, and it could decay immediately. It could decay 1000 years from now. It is like saying a flipped coin will come up heads or tail—you cannot predict the outcome of each throw. Over time, though, the mean probability of getting a “head” or a “tail” will approach 50% as you include increasing number of coin tosses.
This answer seems to indicate that probability is useless (and half life is a probabilistic number), but it is not. If you started adding to the same trap the same radioactive atoms, the uncertainty in how many atoms would decay in a particular time would decrease. As your collection increases, the progression of decay would follow the half-life curve increasingly closer. If you grab one atom out of the mix, however, you will be back to the beginning: you will not be able to predict how long that atom has left before it decays (assuming it has not yet). With a very large collection of radioactive atoms, your equation becomes very precise. As the collection of radioactive atoms decreases, the equation becomes less precise.
This phenomenon shows some important facts about probability. First, consider the decay of carbon-14 (about 5700 years). It is used to date bones, but it is accurate only to a point. As a number of half life cycles has passed, the amount of carbon-14 left is small, and the uncertainty in the decay curve grows. Generally, about 10 half life cycles is about the maximum used to date bones. Thus, it is possible to date bones to about 57,000 years. Beyond that, other methods besides C14 dating need to be used. If someone says they have carbon dated a dinosaur, you know their answer is wrong! However, you could say with certainty that the dinosaur bone is older than 57,000 years because the carbon-14 signature would be gone.
The 50% probability is the probability that the particle will NOT decay at some time during the first half-life. IF you do not look at the particle until one half-life has passed, the particle has a 50% chance of still being there. If the particle is still there, wait until a second half-life has passed. The particle has a 50% chance of remaining again.
If you should wait for two-half-lives before looking, then the particle will have only a 25% chance of still being there. The probability that the particle will decay at an exact time is too small to measure. Probability only means something when time exists for an event to occur. You can ask, “What is the probability that something will happen during this time period?” You can ask, “What is the probability that something will happen before or after a specific time?” It means nothing to ask, “What is the probability that something will happen at exactly a specific time?”
Dr. Ken Mellendorf
Illinois Central College
I think you are caught in a confusion about the concept/definition of PROBABILITY. Probability is a statistical concept. It only has meaning for a large number of particles/events. It is by definition an average. Looking at a single atom gives you zero information about when/if that particular atom will undergo radioactive decay. It gives you no information about single atoms.
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Update: June 2012