I am not sure about my answer. In particular, part b of the following question.
Pizza orders arrive according to a Poisson process of rate 20 per hour. Orders are independently for a vegetarian pizza with probability 1/ 4 , and for a meat pizza with probability 3/4
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a. Six orders arrived between 6:45pm and 7:00pm. Given this, what is the probability that fourteen orders arrive between 7:00pm and 7:45pm?
My Attempt: Let $t=$ time in min. So $\lambda$ per mininute=$20/60=1/3$
$P(N(60)-N(15)=14|N(15)=6)=P(N(45)=14)=\frac{(1/3*45)^{14}}{14!} exp(-1/3*45)$
b. During a particular 60 minute period, 4 vegetarian orders were received. What is the probability that all 4 of them came during the first 30 minutes?
My Attempt:
The conditional poisson would be uniform distributed between (0,60).
Therefore, $P(orders\leq30min)=(30/60)^4=1/16$