Question

This problem is inspired by the real-life search for Air France Flight 447. After the plane vanished in 2009, initial searches failed, leaving a seemingly impossible task. The breakthrough came from applying Bayes' theorem, which treats a failed search not as a dead end, but as valuable new evidence. By formally updating their beliefs about the plane's location after each unsuccessful search, investigators dramatically narrowed the possibilities and found the wreckage. This problem lets you apply the same powerful logic. An aircraft has gone missing over the ocean. Based on the last known data, experts have defined three possible search zones with the following prior probabilities of containing the wreckage: • Zone A: $P(\text{Location} = A) = 0.60$ • Zone B: $P(\text{Location} = B) = 0.30$ • Zone C: $P(\text{Location} = C) = 0.10$ The search equipment is effective, but not perfect. The probability of finding the wreckage if it is in the zone being searched is 80%. Therefore, the probability of failing to find the wreckage in a zone where it is located is 20%. A full search of Zone A is completed, and no wreckage is found. (a) The new evidence, E, is explicitly defined as: the outcome of the search mission in Zone A, which resulted in no wreckage being found. Given this evidence, state the conditional probabilities of E occurring for each possible location. Specifically, what are $P(E|\text{Location} = A)$, $P(E|\text{Location} = B)$, and $P(E|\text{Location} = C)$? (b) Using the law of total probability, calculate the overall probability of the search in Zone A failing, $P(E)$. (c) Using Bayes' Rule, calculate the updated (posterior) probabilities for the wreckage being in each of the three zones, given the failed search in Zone A: $P(\text{Location} = A|E)$, $P(\text{Location} = B|E)$, and $P(\text{Location} = C|E)$. Which zone is now the most probable location for the wreckage?

Name: this problem is inspired by the real life search for air france flight 447 after the plane vanished in 2009 initial searches failed leaving a seemingly impossible task the breakthrough came 40161
Uploaded: 2022-09-16T19:01:45-08:00
Duration: 4 min 16 s
Channel: Kari Hasz
Description: this problem is inspired by the real life search for air france flight 447 after the plane vanished in 2009 initial searches failed leaving a seemingly impossible task the breakthrough came 40161

          This problem is inspired by the real-life search for Air France Flight 447. After the plane vanished in 2009, initial searches failed, leaving a seemingly impossible task. The breakthrough came from applying Bayes' theorem, which treats a failed search not as a dead end, but as valuable new evidence. By formally updating their beliefs about the plane's location after each unsuccessful search, investigators dramatically narrowed the possibilities and found the wreckage. This problem lets you apply the same powerful logic.
An aircraft has gone missing over the ocean. Based on the last known data, experts have defined three possible search zones with the following prior probabilities of containing the wreckage:

• Zone A: $P(\text{Location} = A) = 0.60$

• Zone B: $P(\text{Location} = B) = 0.30$

• Zone C: $P(\text{Location} = C) = 0.10$

The search equipment is effective, but not perfect. The probability of finding the wreckage if it is in the zone being searched is 80%. Therefore, the probability of failing to find the wreckage in a zone where it is located is 20%.

A full search of Zone A is completed, and no wreckage is found.

(a) The new evidence, E, is explicitly defined as: the outcome of the search mission in Zone A, which resulted in no wreckage being found. Given this evidence, state the conditional probabilities of E occurring for each possible location. Specifically, what are $P(E|\text{Location} = A)$, $P(E|\text{Location} = B)$, and $P(E|\text{Location} = C)$?

(b) Using the law of total probability, calculate the overall probability of the search in Zone A failing, $P(E)$.

(c) Using Bayes' Rule, calculate the updated (posterior) probabilities for the wreckage being in each of the three zones, given the failed search in Zone A: $P(\text{Location} = A|E)$, $P(\text{Location} = B|E)$, and $P(\text{Location} = C|E)$. Which zone is now the most probable location for the wreckage?

this problem is inspired by the real life search for air france flight 447 after the plane vanished in 2009 initial searches failed leaving a seemingly impossible task the breakthrough came 40161

Added by Amber J.

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