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Pizza orders arrive according to a Poisson process of rate $20$ per hour. Orders are independently for a vegetarian pizza with probability $\frac14$ , and for a meat pizza with probability $\frac34$.

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?

What I tried: I tried this through binomial: $\binom{20}{14} (0.75)^{14} (0.25)^6$.

Is this correct?

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?

Why is the answer (30/60)^4 = 1/16

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Guide:

  • The question is not asking about vegetarian or meat pizza, there is no reason to use all the numbers that is given to you.

  • Note that the number of orders between 6:45pm and 7:00pm is independent from the number of orders between 7:00pm and 7:45pm.

  • The number of orders between 7:00pm and 7:45pm follows a Poisson distribution, try to find the rate of the Poisson distribution.

  • Note that for Binomial distribution, the setting is there are $n$ events and you want to know how many of them are successful.

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  • $\begingroup$ By poisson and combinations method: I took the probability of (6 orders in 15 minutes)* (14 orders in 45 minutes). The entire thing divided by getting 20 order in a hour. $\endgroup$ – user218970 Sep 9 '18 at 10:51
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    $\begingroup$ The mistake I believe I have made is taking into consideration all the other stuff. It should simply be 14 orders in 45 minutes, isn't it? $\endgroup$ – user218970 Sep 9 '18 at 10:52
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    $\begingroup$ $6$ orders in $15$ minutes is irrelevant. It is possible to have more than $21$ orders in $45$ minutes too. also, note that the question is asking about probability. $\endgroup$ – Siong Thye Goh Sep 9 '18 at 10:54
  • $\begingroup$ If say, it was chronologically in a different order, it would be relevant right? Like, suppose, Probability that 14 orders arrive between 6:45 - 7:30 given that there were 6 orders between 7:30 - 7:45? $\endgroup$ – user218970 Sep 9 '18 at 10:58
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    $\begingroup$ For poisson process, events happening disjoint time intervals are independent, it is still irrelevant. $\endgroup$ – Siong Thye Goh Sep 9 '18 at 10:59

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