Math Problem Statement
Solution
This problem involves Bayesian inference, which aims to update our belief about which hypothesis (in this case, search area) is most likely correct based on prior probabilities and new evidence (the plane's search results). Here's a breakdown of how we can approach this using Bayes' Theorem.
Problem Setup:
- We have three areas to search: , , and .
- Prior probabilities (before any search evidence):
- The likelihood of spotting the hiker given they are in a specific area:
- For area , there's a 50% chance the plane will spot the hiker if they are there:
- For area , the plane has a 30% chance of spotting the hiker:
- For area , the plane has a 20% chance of spotting the hiker:
- For area , there's a 50% chance the plane will spot the hiker if they are there:
Question:
We are asked which area ground-based rescue teams should search first, which involves determining the posterior probability (i.e., the probability that the hiker is in an area, given the search result).
Step-by-Step Bayesian Calculation:
We apply Bayes' Theorem to each area. The formula for the posterior probability is:
Where:
- is the prior probability of the hiker being in area ,
- is the likelihood of spotting the hiker in area ,
- is the total probability of the event (in this case, spotting the hiker).
To compute , we sum over all areas:
Substituting the values:
Posterior Probabilities:
Now we calculate the posterior probability for each area.
-
For Area :
-
For Area :
-
For Area :
Conclusion:
The areas and have the same posterior probability of 0.353, which is higher than that for area . Therefore, rescue teams should first search in areas and , prioritizing these areas equally.
Do you have any questions, or would you like further details on any part of this explanation?
Related Questions:
- How can we adjust the prior probabilities if new information becomes available?
- What would happen if the likelihood of spotting the hiker in area were changed?
- How would the search strategy change if the plane could continue searching for longer periods?
- What if the areas had different terrain difficulties for ground-based rescue teams?
- How does the fog mentioned in the problem affect our Bayesian calculations?
Tip:
In Bayesian problems, always start by carefully identifying the prior probabilities and likelihoods, as these are critical for updating your beliefs.
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Math Problem Analysis
Mathematical Concepts
Bayesian Inference
Probability
Posterior Probability
Formulas
P(Hi | E) = (P(E | Hi) * P(Hi)) / P(E)
P(E) = P(E | A)P(A) + P(E | B)P(B) + P(E | C)P(C)
Theorems
Bayes' Theorem
Suitable Grade Level
Undergraduate Level (Statistics/Probability)
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