Math Problem Statement

Suppose a hiker is lost! There are three search areas A, B, C, initially considered likely as follows: P(A) = 1/6, P(B) = 1/3, P(C) = 1/2. A plane will conduct a search. Area A is open, and the pilot estimates they have a 50:50 chance of spotting the hiker if they’re in area A. Areas B and C are wooded, with odds 30:70 and 20:80, respectively. The plane does not spot the hiker in area B before fog sets in, halting its search. Which area should ground-based rescue teams search first?

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: AA, BB, and CC.
  • Prior probabilities (before any search evidence):
    P(A)=16,P(B)=13,P(C)=12P(A) = \frac{1}{6}, \quad P(B) = \frac{1}{3}, \quad P(C) = \frac{1}{2}
  • The likelihood of spotting the hiker given they are in a specific area:
    • For area AA, there's a 50% chance the plane will spot the hiker if they are there:
      P(Spot | A)=0.5P(\text{Spot | A}) = 0.5
    • For area BB, the plane has a 30% chance of spotting the hiker:
      P(Spot | B)=0.3P(\text{Spot | B}) = 0.3
    • For area CC, the plane has a 20% chance of spotting the hiker:
      P(Spot | C)=0.2P(\text{Spot | C}) = 0.2

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:

P(HiE)=P(EHi)P(Hi)P(E)P(H_i | E) = \frac{P(E | H_i) P(H_i)}{P(E)} Where:

  • P(Hi)P(H_i) is the prior probability of the hiker being in area ii,
  • P(EHi)P(E | H_i) is the likelihood of spotting the hiker in area ii,
  • P(E)P(E) is the total probability of the event (in this case, spotting the hiker).

To compute P(E)P(E), we sum over all areas: P(E)=P(EA)P(A)+P(EB)P(B)+P(EC)P(C)P(E) = P(E | A)P(A) + P(E | B)P(B) + P(E | C)P(C)

Substituting the values: P(E)=(0.5×16)+(0.3×13)+(0.2×12)P(E) = (0.5 \times \frac{1}{6}) + (0.3 \times \frac{1}{3}) + (0.2 \times \frac{1}{2}) P(E)=112+110+110P(E) = \frac{1}{12} + \frac{1}{10} + \frac{1}{10} P(E)=0.0833+0.1+0.1=0.2833P(E) = 0.0833 + 0.1 + 0.1 = 0.2833

Posterior Probabilities:

Now we calculate the posterior probability for each area.

  1. For Area AA: P(AE)=P(EA)P(A)P(E)=0.5×160.2833=112×10.2833=0.294P(A | E) = \frac{P(E | A) P(A)}{P(E)} = \frac{0.5 \times \frac{1}{6}}{0.2833} = \frac{1}{12} \times \frac{1}{0.2833} = 0.294

  2. For Area BB: P(BE)=P(EB)P(B)P(E)=0.3×130.2833=110×10.2833=0.353P(B | E) = \frac{P(E | B) P(B)}{P(E)} = \frac{0.3 \times \frac{1}{3}}{0.2833} = \frac{1}{10} \times \frac{1}{0.2833} = 0.353

  3. For Area CC: P(CE)=P(EC)P(C)P(E)=0.2×120.2833=110×10.2833=0.353P(C | E) = \frac{P(E | C) P(C)}{P(E)} = \frac{0.2 \times \frac{1}{2}}{0.2833} = \frac{1}{10} \times \frac{1}{0.2833} = 0.353

Conclusion:

The areas BB and CC have the same posterior probability of 0.353, which is higher than that for area AA. Therefore, rescue teams should first search in areas BB and CC, prioritizing these areas equally.


Do you have any questions, or would you like further details on any part of this explanation?

Related Questions:

  1. How can we adjust the prior probabilities if new information becomes available?
  2. What would happen if the likelihood of spotting the hiker in area AA were changed?
  3. How would the search strategy change if the plane could continue searching for longer periods?
  4. What if the areas had different terrain difficulties for ground-based rescue teams?
  5. 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)