Exponential Distribution: Calculating the Half-Life & Survival Rate of a Rock"

[Bachelier]

A piece of rock contains 10^20 atoms of a particular substance. Each atom has an expoentially distributed lifetime with a half-life of one century. How many centurites must pass before

there is about a 50% chance that at least one atom remains. What assumptions are you making?

answer:

so P (at least one survives past t) = P (no one does) = .5

now, I'm making the assumption that Prob of survival is so small and since n is huge, this follows a poisson disn.

thus .5 = P(k=0) = e^-μ

then μ = ln 5

now μ = np = 10²⁰* e^{-ln 2 t}. ln 2 is my parameter since half time is 1 century.

thus t = ln(10²⁰/ln 5) * (ln 2)^-1 ≈ 65 years

 

[mathman]

Your time unit is centuries, not years. So your answer (I didn't check arithmetic) is 65 centuries.

 

[Bachelier]

what about the theory, is it correct?

 

[mathman]

.5 = P(k=0) = e^-μ

then μ = ln 5

Above has error, μ = -ln.5 = ln2

The general idea is correct. You might try a binomial to check. The result should be about the same.

 

[blue_raver22]

So, after approximately 65 years, there is a 50% chance that at least one atom will remain in the rock. This calculation assumes that the atoms have a constant and independent probability of decaying, and that there are no external factors affecting the decay process. It also assumes that the distribution of lifetimes of the atoms follows an exponential distribution.