Find the probability that no rock noise will be recorded for at least $30$ minutes.

Question

Rock noise in an underground mine occurs at an average rate of three per hour. Find the probability that no rock noise will be recorded for at least 30 minutes.

I was able to get the density function for it as $f(x)=3e^{-3x}.$$ How do I get the probability?

Is this a Poisson Distribution or Exponential?

Answer 1

If the average is $3$ per hour, then the time to the first arrival has density $$f_X(x) = 3\exp\left\{-3x\right\}.$$

Further, the question asks $P(X> 30\text{ min}) = P(X> .5\text{ hr})$. Compute this using the usually integration.

Warning!

This is correct if you logic is that the time to the first arrival follows an exponential distribution. Otherwise, recognize that you are dealing with a Poisson distribution.

In other words,

$$A = \{\text{First arrival after 30 mins}\}\iff B =\{\text{Number of arrivals in 30 mins is zero}\}.$$

Hence $$P(A) = P(B) = e^{-\mu t}\frac{\mu^0}{0!} = e^{-3(.5)} = e^{-1.5}=0.2231302.$$

Answer 2

Use the Poisson distribution with $\lambda = 1.5$ for a half hour period.

Then, it's just $P(0) = e^{-1.5}$