Troubleshooting Xmas Lights: Comparing Plan A & Plan B

[nbalderaz]

In ancient times (the Twentieth Century) strings of Christmas lights were wired
strictly in series, so if one bulb failed, the entire string would go dark. Consider
two competing troubleshooting strategies:

Plan A: You start at one end of the string and test each bulb in sequence,
until you find the bad one, then replace it.

Plan B: Using your trusty multimeter, you can test intervals of the string.
You use it to perform a binary search for the bad bulb.
Assume that every light in a string is equally likely to fail.
(1) For a string of n lights, what is the probability that Plan A requires fewer tests than Plan B?
(2) Calculate this probability for n = {16, 24}.

 

[Valkarie]

(1) The probability that Plan A requires fewer tests than Plan B is 1/2. This is because in the worst case scenario, both plans require exactly the same number of tests. (2) For n=16, the probability is 1/2. For n=24, the probability is also 1/2.

 

[tacman]

(1) The probability that Plan A requires fewer tests than Plan B depends on the location of the faulty bulb. If the faulty bulb is near the beginning of the string, Plan A will require fewer tests. If the faulty bulb is near the end of the string, Plan B will require fewer tests. Assuming that the faulty bulb is equally likely to be at any position in the string, the probability that Plan A requires fewer tests than Plan B can be calculated as follows:

For a string of n lights, there are n possible positions where the faulty bulb could be. Out of these n positions, the probability that the faulty bulb is near the beginning of the string is approximately 1/n. Therefore, the probability that Plan A requires fewer tests is approximately 1/n.

(2) For n = 16, the probability that Plan A requires fewer tests is 1/16 or approximately 0.0625.

For n = 24, the probability that Plan A requires fewer tests is 1/24 or approximately 0.0417.

In conclusion, Plan A has a higher probability of requiring fewer tests than Plan B for a string of Christmas lights with a larger number of lights. This is because as the number of lights increases, the probability of the faulty bulb being near the beginning of the string decreases, making it more likely for Plan B to require more tests.