Plutonium Decay
An atom of Plutonium decays at an average rate of one decay per second. What is the probability that it does not decay in the next 3 seconds? Express your answer in the form \(e^{a}\). Find the value of \(a\).
Answer
Let \(X\) be the time until the atom decays. Then \(X\) follows an exponential distribution with rate \(\lambda = 1\) (i.e. mean \(1\) second). The probability that the atom does not decay in the next 3 seconds is
\[
P(X > 3) \;=\; e^{-1\cdot 3} \;=\; e^{-3}.
\]
Thus, in the form \(e^{a}\), we have \(a = -3\).