The probability of no cars passing within a certain time interval Let's say there is an induction loop in a road capable of counting the number of cars passing over it. By keeping a list of moments for when a car passed the detection loop, I am able to determine the average interval $\mu$ and variance  $\sigma^2$ there over. I also know the amount of intervals $N$ measured and the last moment a car passed the induction loop.

I wish to know the chance of no cars passing for a time interval since the last moment a car passed the loop. This question seems to ask for a Poisson-distribution, but these seem no to keep into account the variance of time intervals, nor do they seem to use the amount of measurements. Which method should I use to determine the probability of no car passing the detection loop within a certain interval?

known values:


$\mu$: The average interval between two cars passing the induction loop
$\sigma^2$: The variance in these intervals
$N$: the amount of interval measurements


unknown:


$P(t,c)$: The chance of $c$ cars passing the induction loop in interval $t$


where:


$t$: the interval for which the probability is sought
$c$: the amount of cars which should pass within the interval. 0 in my case.
Contrary to the existing answer, the probability does depend on the variance.

If you kept the entire information from the previous measurements, you could estimate the cumulative distribution function $F(x)$ of the time interval between car passages using the empirical distribution function. Since you only recorded the first two moments, you can approximate it using a normal distribution. The cumulative distribution function of a normal distribution with mean $\mu$ and variance $\sigma^2$ is

$$
F(x)=\frac{1}{2}\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right]\;.
$$

Given $F(x)$, the probability that no cars will pass during the interval $t$ is $1-F(t)$.
Variance is not needed, and it is perfectly fine to use Poisson formula. So your answer would be just

$$P(X=0)=e^{-\mu}$$