# Exponential dsn. please check my work

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

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

## Answers and Replies

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

what about the theory, is it correct?

mathman
Science Advisor
.5 = P(k=0) = e

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.