Locating missing person (Bayes Theorem + Conditional Probability?)

Question

So I came across a question that is kinda interesting. Here's the question:

Imagine we have 2 locator devices to find a missing person in the mountains. The probability that the missing person detected by device A is 0.8. Similarly, probability that the missing person detected by device B is 0.9.

But a single locator device is unable to locate the exact location of the person. In order to get accurate coordinates of the missing person, he must be detected by both locator devices.

Based on device A alone, the probability of detecting the exact coordinates of the missing person is 0.7 and similarly for device B with probability of 0.4

Find the probability of locating the missing person?

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Attempt:

$P(A)= 0.8$ -> probability missing person detected by device A

$P(B)= 0.9$ -> probability missing person detected by device B

$P(C)= 0.7$ -> probability device A accurately locating missing person.

$P(D) = 0.4$ -> probability device B accurately locating missing preson.

$P(A \cap C) = 0.8 \cdot 0.7$? -> Probability missing person detected by device A with accurate coordinates/position. Likewise for device B?

It's been a year since I last touched probability. Any idea on how to approach this problem?

Answer 1

It is not clear to me (a) whether (given B fails to detect) device A accurately locates the missing person with probability $0.8 \times 0.7$ rather than just $0.7$, (b) whether the two devices act independently in detection, and (c) whether both devices detecting the missing person guarantees accurate location.

If the answer to all three is yes, then you could approach this by looking at the cases:

• Both detect the missing person with probability $0.8\times 0.9$. Given that, the probability of accurate location is $1$

• A detects the missing person but B does not with probability $0.8 \times 0.1$. Given that, the probability of accurate location is $0.7$

• B detects the missing person but A does not with probability $0.2 \times 0.7$. Given that, the probability of accurate location is $0.4$

• Neither detect the missing person with probability $0.8\times 0.9$. Given that, the probability of accurate location is $0$

That gives an overall probability of an accurate location of $$(0.8\times 0.9 \times 1 )+(0.8\times 0.1 \times 0.7 )+(0.2\times 0.9 \times 0.4 )+(0.2\times 0.1 \times 0 )$$

Answer 2

Alternative assumptions: (a) each machine detects independently of the other with the given probabilities; (b) they both need to detect for any chance of location; (c) given they both detect, each machine can independently locate accurately with the given probabilities; (d) they must both locate accurately for the missing person to be found. Then

The probability they both detect is $0.8 \times 0.9$

If they both detect then the conditional probability they both locate accurately is $0.7 \times 0.4$

So the overall probability of a double detection and double accurate location is

$$0.8 \times 0.9 \times 0.7 \times 0.4$$