The atomic nucleus of plutonium-239 (239Pu) is unstable, and it spontaneously decays to form uranium-235 (235U):

This is the first step in the radioactive decay chain of nuclear reactors. 239Pu −→ 235U.

The time for an atom to decay can be modelled by the Exponential(λ) distribution, where the rate of decay is λ = 2.86 × 10−5 year−1 . Let X ∼ Exponential(2.86 × 10−5 ) be the time taken for an atom to decay.

a. What is the probability that an atom of 239Pu will decay in the next 1000 years?

b. What is the probability that an atom of 239Pu will not decay for the next 2000 years?

c. The half-life, t1/2, is the time taken for half of the plutonium atoms to decay. This means that t1/2 satisfies P(X ≤ t1/2) = 0.5.
Show that t1/2 = − ln(0.5)
λ
, and hence find the half-life of the 239Pu decay process.
The half-life measures how long the radioactive plutonium atoms persist in the environment. As you can see, they persist for a very long time.