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The lifetime, in years, of a Carbon-14 atom can be modeled by a random variable \(X\) following an exponential distribution with parameter \(\lambda\).
The
half-life
of this atom is the real number \(t\) such that the probability that it disintegrates before \(t\) years is equal to \(\frac{1}{2}\).
We know that the half-life of Carbon-14 is equal to 5730 years.
Calculate the parameter \(\lambda\) of the distribution modeling the lifetime \(X\) of Carbon-14 (give the result in scientific notation rounded to 2 decimal places).
In the following, we will take \(\lambda = 12 \times 10^{-5}\). Calculate the probability that a Carbon-14 atom disintegrates before 2000 years (round to three decimal places).
What is the probability that the lifetime of Carbon-14 is greater than two half-lives?
Determine the value of \(x\) such that \(P(X > x) = 0.01\). (Round to the nearest year). Interpret this result.
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