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Answer :
Final answer:
Therefore, the probability that there is actually a gas leak when the sniffer detects one is approximately 4.72%.
Explanation:
To determine the probability of an actual gas leak if the sniffer detects a leak, we can apply Bayes' theorem, which relates the conditional and marginal probabilities of stochastic events. Given that the sensitivity (true positive rate) of the sniffer is 0.99 and the specificity (true negative rate) is 0.98, and the prior probability of there being a gas leak is 0.001, we need to calculate the posterior probability.
The formula for the posterior probability (P(A|B)) in the context of our problem is:
P(A|B) = (P(B|A) * P(A)) / [(P(B|A) * P(A)) + (P(B|not A) * P(not A))]
Where:
- P(A) is the prior probability of a gas leak (0.001)
- P(B|A) is the sensitivity or the probability of a positive test given a gas leak (0.99)
- P(not A) is the probability of there being no gas leak, which is 1 - P(A) (0.999)
- P(B|not A) is the probability of a false positive, which is 1 - specificity (0.02)
Plugging in the values:
P(A|B) = (0.99 * 0.001) / [(0.99 * 0.001) + (0.02 * 0.999)]
P(A|B) ≈ 0.0472
Therefore, the probability that there is actually a gas leak when the sniffer detects one is approximately 4.72%.
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