Let (X) denote the time between detections of a particle with a Geiger counter and assume that

Question:

Let $X$ denote the time between detections of a particle with a Geiger counter and assume that $X$ has an exponential distribution with $E(X)=1.4$ minutes. The probability that we detect a particle within 0.5 minute of starting the counter is

\[ P(X<0.5)=F(0.5)=1-e^{-0.5 / 1.4}=0.30 \]

Now, suppose that we turn on the Geiger counter and wait 3 minutes without detecting a particle. What is the probability that a particle is detected in the next 30 seconds?

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