Fairy Lights

Dec 2019
52
1
ok
The production of fairy lights at the North Pole goes on very smoothly. The elves Dincho, Antonia, and Bogdan are very happy with their newly invented machine, which simplifies their work tremendously. Therefore, they invite all the other elves to a board game party.
Unfortunately, they do not look after the production anymore and the Grinch is able to sneak into the plant. He sabotages the new machine and additionally puts broken light bulbs, that were sorted out already, back into the supply of functioning light bulbs. Elf Sinan notices this at first, and subsequently all elves are very upset: what will the elf-in-chief, Nicolas, say if the fairy lights cannot be shipped on time? Immediately, the elves Daria and Thore begin to fix the machine. Elves Bendix and Jan start to test the light bulbs again and sort them accordingly.
But time is running. Elf Joana says, “Testing every light bulbs individually costs too much time. We should let the machine put together the fairy lights and test the assembled fairy lights afterwards.”
Elf Lennart agrees, “The light bulbs are broken with the small probability of 0.1 %—accordingly, many of the fairy lights will work if we just assemble them.”
Elf Kai decides, “We put together the fairy lights and then test the assembled fairy lights. Afterwards, we individually test all light bulbs of all the fairy lights that do not work. Finally, we exchange the broken light bulbs by functioning ones, that have already been checked by Bendix and Jan.”
Elf Phillipp takes a look at the schedule, “For the next shipment, we need 100 fairy lights consisting of 100 light bulbs each.”
Testing expert Sara adds, &rldquo;For each test—regardless of assembled fairy lights or individual light bulb—we need 3 seconds.”

Now, everyone is hard at work. Meanwhile, the mathematics-loving elves amuse themselves by predicting the test outcomes. However, one elf is mistaken. Which one?
  1. Elf Taha says, “If we are lucky, then we are finished with testing after exactly 5 minutes.”
  2. Elf Marwan replies, “If we have bad luck, then we need 8 hours and 25 minutes for testing.”
  3. Elf Lara points out, ”In the worst case—that is, if all of the fairy lights are broken—we would need 5 minutes less, when testing all light bulbs individually.”
  4. Elf Andi replies, “But a single one of the fairy lights is fully functioning with a probability of over 90 %.”
  5. Elf Katharina adds, “However, the best case—that is, if all the fairy lights are working—occurs only with a probability of less then 0.005 %.”
  6. Elf Jonathan indicates that, “The worst case—i. e., that all fairy lights are broken—occurs with a much smaller probability of less than 10-200 %.”
  7. Elf Yura says, “Anyway, the probability of exactly two of the fairy lights being broken is not equal to the probability of exactly three of the fairy lights being broken.”
  8. Elf Isabella adds, “The probability of exactly five of the fairy lights being broken is less than 5 %.”
  9. Elf Fabian points out, “We can expect that less than eleven of the hundred fairy lights are broken.”
  10. Elf Clemens concludes, “Hence, we can expect that we need less than an hour for testing all hundred of the fairy lights.”
 
Dec 2019
52
1
ok
The production of fairy lights at the North Pole goes on very smoothly. The elves Dincho, Antonia, and Bogdan are very happy with their newly invented machine, which simplifies their work tremendously. Therefore, they invite all the other elves to a board game party.

Unfortunately, they do not look after the production anymore and the Grinch is able to sneak into the plant. He sabotages the new machine and additionally puts broken light bulbs, that were sorted out already, back into the supply of functioning light bulbs. Elf Sinan notices this at first, and subsequently all elves are very upset: what will the elf-in-chief, Nicolas, say if the fairy lights cannot be shipped on time? Immediately, the elves Daria and Thore begin to fix the machine. Elves Bendix and Jan start to test the light bulbs again and sort them accordingly.

But time is running. Elf Joana says, “Testing every light bulbs individually costs too much time. We should let the machine put together the fairy lights and test the assembled fairy lights afterwards.”
Elf Lennart agrees, “The light bulbs are broken with the small probability of 0.1 %—accordingly, many of the fairy lights will work if we just assemble them.”
Elf Kai decides, “We put together the fairy lights and then test the assembled fairy lights. Afterwards, we individually test all light bulbs of all the fairy lights that do not work. Finally, we exchange the broken light bulbs by functioning ones, that have already been checked by Bendix and Jan.”
Elf Phillipp takes a look at the schedule, “For the next shipment, we need 100 fairy lights consisting of 100 light bulbs each.”
Testing expert Sara adds, &rldquo;For each test—regardless of assembled fairy lights or individual light bulb—we need 3 seconds.”

Now, everyone is hard at work. Meanwhile, the mathematics-loving elves amuse themselves by predicting the test outcomes. However, one elf is mistaken. Which one?

  1. Elf Taha says, “If we are lucky, then we are finished with testing after exactly 5 minutes.”
  2. Elf Marwan replies, “If we have bad luck, then we need 8 hours and 25 minutes for testing.”
  3. Elf Lara points out, ”In the worst case—that is, if all of the fairy lights are broken—we would need 5 minutes less, when testing all light bulbs individually.”
  4. Elf Andi replies, “But a single one of the fairy lights is fully functioning with a probability of over 90 %.”
  5. Elf Katharina adds, “However, the best case—that is, if all the fairy lights are working—occurs only with a probability of less then 0.005 %.”
  6. Elf Jonathan indicates that, “The worst case—i. e., that all fairy lights are broken—occurs with a much smaller probability of less than 10-200 %.”
  7. Elf Yura says, “Anyway, the probability of exactly two of the fairy lights being broken is not equal to the probability of exactly three of the fairy lights being broken.”
  8. Elf Isabella adds, “The probability of exactly five of the fairy lights being broken is less than 5 %.”
  9. Elf Fabian points out, “We can expect that less than eleven of the hundred fairy lights are broken.”
  10. Elf Clemens concludes, “Hence, we can expect that we need less than an hour for testing all hundred of the fairy lights.”

Does anyone know how to solve this? :(