Since carbon is fundamental to life, occurring along with hydrogen in all organic compounds, the detection of such an isotope might form the basis for a method to establish the age of ancient materials.
During the lifetime of an organism, the amount of c14 in the tissues remains at an equilibrium since the loss (through radioactive decay) is balanced by the gain (through uptake via photosynthesis or consumption of organically fixed carbon).
However, when the organism dies, the amount of c14 declines such that the longer the time since death the lower the levels of c14 in organic tissue.
Working with several collaboraters, Libby established the natural occurrence of radiocarbon by detecting its radioactivity in methane from the Baltimore sewer.
In contrast, methane made from petroleum products had no measurable radioactivity.
Scientists look at half-life decay rates of radioactive isotopes to estimate when a particular atom might decay.
A useful application of half-lives is radioactive dating.This has to do with figuring out the age of ancient things.If you could watch a single atom of a radioactive isotope, U-238, for example, you wouldn’t be able to predict when that particular atom might decay.Since that decays and the other $\ce$-isotopes don't, over the years, the percentage of $^\ce$ is getting lower and lower at a constant rate (you can calculate that rate with the half-time of $^\ce$).Now when you measure the relative amount of $^\ce$ in a skeleton, you know since when it hasn't eaten anymore, so you know how old it is.The ensuing atomic interactions create a steady supply of c14 that rapidly diffuses throughout the atmosphere.