Question

The half-life of Carbon-14 is 5,730 years. What is the probability that a randomly selected Carbon-14 atom will not yet have decayed in 4,000 years’ time?

Answer 1

here as life of an atom follows exponential distribution with parameter $\beta$ =mean life =half life/ln(2)=5730/ln(2)

=8266.643 years

hence   probability that a randomly selected Carbon-14 atom will not yet have decayed in 4,000 years’ time

=P(X>4000) =e^{-x/ $\beta$} =e^{-4000/8266.643}=0.6164