This is for homework, but I'm stuck cause I can't find something like this on my book, or in the internet for that matter.
Say I have a restaurant that's open the 24 hours, and we know that costumers enter the establishment according a Poisson process at a rate of 5 costumers/hour.
1. Given that 6 costumers arrived from 1:00 am and 2:30 am, what's the probability that less than 3 customers arrive from 2:30 am to 4:00 am?
I have modeled this question like this: P(C<3|X=6), being C the amount of customers that could arrive and X the customers that have already arrived. so I use the definition of conditional probability: $\frac{P(C<3\bigcap X=6)}{P(X=6)}$
I ended up using this property that seems shady: $P(A\bigcap B)=P(A) P(B)$
Make the math and I end up with $\frac{0.1367 \cdot 0.059}{0.1367}=0.059$ which seems reasonable, but I don't think is right. So help me here if it's not ok.
2. If we know that 30 customers arrived from 10:00 pm to 4:00 am, what's the probability that 20 customers arrived from 1:30 am to 3:15 am?
This questions has me stumped. I think I should break the times in something like: $[10:00, 1:30)\bigcup [1:30, 3:15]\bigcup (3:15, 4:00]$
And then do the probabilities separate, and add them up at the end, but that seems wrong, and I don't know what to do...
Please help