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, #100 You 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 is P(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 → ∞, wha can you say about P (N = n)? 9) Using your answer to Q1.8 - find the smallest nmax such that P(N < nmax) = 0.95, i.e., find an upper bound that you are 95% confident the true number of drones is less than. • Bonus 1: How many more drones with serial numbers less than #700 would you have to observe before being 95% confident that there were at most 1000 drones? • Bonus 2: What assumptions in this analysis aren’t necessarily accurate, and how could they be im- proved? Hint: 1.4 asks you to look at your answer to 1.3 and think, what number of drones would make this sequence of 5 drones -as likely as possible-? Mechanically it asks, what value of N makes your answer to 1.3 as large as possible? 1.1 is relevant here - why? 1.5 asks - hypothetically, if the number of drones were very very large, is this data reasonable or unreasonable to expect? (1.6 asks you to effectively generalize 1.1, 1.2, 1.3, and 1.4 for a general set of serial number observations n_1, n_2, ..., n_k) 1.7, 1.8, 1.9 take the first part of the problem and turn it on its head, asking you - what do you know about N? 1.7 asks, effectively "given the serial numbers observed, how likely is it that there are only 500 drones out in the world?" 1.8 asks you to imagine that the probability of there being n drones 'goes like' the likelihood of seeing these 5 serial numbers if there are actually n drones in the world. i.e., if F(N) is your answer to 1.3, 1.8 asks you to imagine P(N = n) = c*F(n), where c is some constant of proportionality you need to work out. 1.9 asks - after 1.8, you should be able to calculate probabilities like P(N = 1000) or P(N = 500000) exactly. These probabilities are quite small however, so what about the probability that P(N <= 1000)? P(N <= 5000)? Can you find some bound n_max where you are 95% certain the number of drones is less than or equal to n_max, i.e., P(N <= n_max) >= 0.95?
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, #100
You 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
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 is P(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
9) Using your answer to Q1.8 - find the smallest nmax such that P(N < nmax) = 0.95, i.e., find an upper bound that you are 95% confident the true number of drones is less than.
• Bonus 1: How many more drones with serial numbers less than #700 would you have to observe before being 95% confident that there were at most 1000 drones?
• Bonus 2: What assumptions in this analysis aren’t necessarily accurate, and how could they be im- proved?
Hint:
1.4 asks you to look at your answer to 1.3 and think, what number of drones would make this sequence of 5 drones -as likely as possible-? Mechanically it asks, what value of N makes your answer to 1.3 as large as possible? 1.1 is relevant here - why?
1.5 asks - hypothetically, if the number of drones were very very large, is this data reasonable or unreasonable to expect?
(1.6 asks you to effectively generalize 1.1, 1.2, 1.3, and 1.4 for a general set of serial number observations n_1, n_2, ..., n_k)
1.7, 1.8, 1.9 take the first part of the problem and turn it on its head, asking you - what do you know about N?
1.7 asks, effectively "given the serial numbers observed, how likely is it that there are only 500 drones out in the world?"
1.8 asks you to imagine that the probability of there being n drones 'goes like' the likelihood of seeing these 5 serial numbers if there are actually n drones in the world. i.e., if F(N) is your answer to 1.3, 1.8 asks you to imagine P(N = n) = c*F(n), where c is some constant of proportionality you need to work out.
1.9 asks - after 1.8, you should be able to calculate probabilities like P(N = 1000) or P(N = 500000) exactly. These probabilities are quite small however, so what about the probability that P(N <= 1000)? P(N <= 5000)? Can you find some bound n_max where you are 95% certain the number of drones is less than or equal to n_max, i.e., P(N <= n_max) >= 0.95?
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