Half-life of 10^20 atoms with exponential lifetime distribution

[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.