Problem 3: Splitting the Atom Having recently gained access to the internet (wikipedia - fantastic! ), after doing some research you've learned three things which you did not know: Radioactive materials steadily decay with a known half life, allowing the prediction of how much radioactive material is left after a given amount of time. This is the basis of techniques like Carbon Dating. An atom of a radioactive isotope decays randomly - it is impossible to predict when it will decay, but in every interval of time, there is some probability of its decay, independent of the previous intervals. This is connected to what's known as Quantum Mechanics. Humans should not be fed radioactive isotopes in their bagels without their knowledge. This is related to the larger subject of Ethics.

This surprises you, however, because the first two statements seem to be in contradiction to one another - if each atom decays randomly, how can there be a well-defined half life, at which point half the radioactive material remains? Shouldn't the half life be random? Couldn't every atom decay all at the same time? You resolve to do some calculation.

Suppose in a scoop of strontium-90, there are N atoms, each labeled atom 1 through atom N . Let Xi(t) be the state of atom i at time t, where Xi(t) = 1 if the atom has not decayed yet, and Xi(t) = 0 if it has. Xi(0) = 1 for all atoms, indicating they all start as not decayed. In each timestep (t t + 1), each undecayed atom has a probability p of decaying. We can define X(t) = X1(t) + X2(t) + X3(t) + . . . + XN (t), which sets X(t) to be the number of undecayed atoms at time t.

1) What is the probability that atom i is undecayed at time t 1, i.e., P (Xi(t) = 1)? (I'm looking for a formula for this probability.)

2) Find (t) = E [X(t)], the expected number of strontium-90 atoms that are left at time t. Show that this decays exponentially, at a rate determined by p. (I'm looking for a formula for (t) that should depend on t.)

3) At what time t is the expected amount of strontium-90 left no more than half the original amount? Call this the half life, t1/2, and show it doesn't depend on the original amount of material. (I'm looking for an expression or formula, t(1/2), which may be in terms of the parameters of the problem. It's worth trying to translate the sentence describing t(1/2) into something more mathematical, to figure out how to solve for t(1/2).)

The previous calculations are meant to show that the expected or average amount of radioactive material left at a given time behaves in a very predictable manner with an exponential decay. However, what about the actual amount of radioactive material?

4) Derive an upper bound on P (X(t) (t)(1 + )) in terms of N , , p, and t. This is the probability that the actual amount of material left at time t is larger than the expected amount by a factor of 1 + . (I'm looking for a formula such that P( X(t) >= mu(t) (1+epsilon) ) is less than or equal to that formula. Note, this probability is obviously less than or equal to 1 - but I'm looking for a bound that is much tighter than that.)

5) Derive an upper bound on P (X(t) (t)(1 )) in terms of N , , p, and t. This is the probability that the actual amount of material left at time t is smaller than the expected amount by a factor of 1 . (Similar to 3.4)

6) Noting the number of atoms in a scoop of strontium-90 is about N 10^23, show that with probability almost 1, X(t1/2) is between 0.999(t1/2) and 1.001(t1/2). (Show that P( 0.999(t(1/2)) <= X(t(1/2)) <= 1.001 (t(1/2)) ) is very, very close to 1.)

As such, the apparent contradiction seems resolved - while it is the expected amount of radioactive material that decays exponentially in a very predictable fashion with a known half life, the actual amount of radioactive material (though random) is concentrated around this with very high probability.

Bonus: All living organisms that we know of contain Carbon. A fraction of this carbon is Carbon-14, a radioactive isotope of Carbon. The proportion of Carbon-14 in an organism is constant over its lifespan, as the organism is constantly replenishing the carbon in its body. However, after the organism dies, the Carbon-14 decays at a known rate, allowing scientists to gauge how long ago the organism was alive based on the amount of Carbon-14 remaining. However, this is not accurate beyond a timescale of about 50, 000 years. Based on the above analysis, why might there be an upper limit on how long a timescale this technique could be applied over?