Practice problem I need help with. This is not an essay writing question. This is a math question. You live a peaceful life in the post-apocalypse, collecting mushrooms, playing with your robot cat, and looking for non-radioactive water. It’s been years since humanity and the Amazon Delivery Drone Swarm wiped each other out, but you still think sometimes, about hiding behind your couch, wondering if the next package you received would be your last. But what few survivors there are haven’t seen any drones, or received any packages, in a very long time.But something has changed. Somewhere, an automated factory seems to have sprung to life, churning out new drones- you’ve seen them zipping through the air at a distance, buzzing angrily as they look for someone to deliver things too. To try to gauge the size of the threat, you’d like to know how many drones there are - but it’s difficult to count them.Watching them through an antique telescope you recovered from the remains of the Museum of Natural History, you note that the drones are stamped with a serial number - surely marking the order they were manufactured? You make note of the serial numbers you see:#743, #238, #71, #912, #100You assume that every drone is stamped with a serial number, probably starting from #1 and going to some number #N, where N is the total number of drones out there. 1) Assuming no gaps in serial numbers, what’s the smallest number of drones that might be out there?

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Practice problem I need help with. This is not an essay writing question. This is a math question.

You live a peaceful life in the post-apocalypse, collecting mushrooms, playing with your robot cat, and looking for non-radioactive water. It’s been years since humanity and the Amazon Delivery Drone Swarm wiped each other out, but you still think sometimes, about hiding behind your couch, wondering if the next package you received would be your last. But what few survivors there are haven’t seen any drones, or received any packages, in a very long time.But something has changed. Somewhere, an automated factory seems to have sprung to life, churning out new drones- you’ve seen them zipping through the air at a distance, buzzing angrily as they look for someone to deliver things too. To try to gauge the size of the threat, you’d like to know how many drones there are - but it’s difficult to count them.Watching them through an antique telescope you recovered from the remains of the Museum of Natural History, you note that the drones are stamped with a serial number - surely marking the order they were manufactured? You make note of the serial numbers you see:#743, #238, #71, #912, #100You assume that every drone is stamped with a serial number, probably starting from #1 and going to some number #N, where N is the total number of drones out there.

1) Assuming no gaps in serial numbers, what’s the smallest number of drones that might be out there?

2) What is the sample space of possible sets of 5-drone observations? How many possible observations are there

3) If all drones have been deployed, and each is equally likely to be observed, what is the probability of observing the specific drones given above?

4) What number of drones would make the probability of observing these specific drones as large as possible?

5) Intuitively, why is it unreasonable to imagine there are ten million drones in the wild?

6) In the general case, if drones{n1, n2, . . . , nk}are observed, where ni is the serial number of the i-th drone observed, what number of drones makes the probability of observing these specific drones as large as possible

The answer to Q1.3 above tells you the number of drones that would make the observed data as likely as possible- but this still might not be the true number of drones. If N is the unknown number of drones, we’d like to know what the probability is that N=n, for any value n.

7) What isP(N= 500)?

8) Assume that if n is a feasible value of N (i.e., it is realistic that there might be n drones, given the observations you’ve made), then P(N=n) is proportional to the probability of observing drones {#743,#238,#71,#912,#100} when there are n total drones. What is the value of P(N=n) as a function of n? As n→ ∞, what can you say about P(N=n)?

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