I have recently tackled this question: If the probability of observing a car in 30 minutes on a highway is 0.95, what is the probability of observing a car in 10 minutes (assuming constant default probability)?
While I fully understand the answer, I was wondering if the proposed answered is a good one? What if you've been asked to calculate the probability of seeing a car in 12 or 13 minutes? This will present a problem with the proposed strategy no?
Is there a way to calculate the probability for any given interval?
Another question would be what does it mean "assuming constant default probability" wouldn't that question follow under the poisson distribution?
The accepted answer gives a good approach. If we model the arrival of cars as a Poisson distribution, the fact that the probability of observing at least one event in $30$ minutes is $0.95$ tells us the probability of observing no events is $0.05$. This gives us that the parameter $\lambda$ of the distribution satisfies $e^{-\lambda}=0.05$ or $\lambda =-\log(0.05) \approx 2.996$ The expected rate of cars to see is then about $\frac {2.996}{30}$ per minute and the expected number to be seen in $12$ minutes is about $\frac {12 \cdot 2.996}{30}=1.198.$ The chance of seeing none in $12$ minutes is then $e^{-1.198}\approx 0.302$