An man was found dead with knife wounds in the biggest family gathering in the world with 4,000 relatives partying in a mansion. The home was put under lockdown so that no guests will leave and fingerprint analysis of the murder weapon using a rapid test kit was positive on the 400th person tested. Assuming that an innocent person has only a 1 in 500 chance of a false positive result, what is the probability that this person is guilty? Select the correct answer below. 0.002 0.125 0.250 Correct Answer 0.111 0.95
Question
An man was found dead with knife wounds in the biggest family gathering in the world with 4,000 relatives partying in a mansion. The home was put under lockdown so that no guests will leave and fingerprint analysis of the murder weapon using a rapid test kit was positive on the 400th person tested. Assuming that an innocent person has only a 1 in 500 chance of a false positive result, what is the probability that this person is guilty? Select the correct answer below. 0.002 0.125 0.250 Correct Answer 0.111 0.95
Solution
The probability that the person is guilty can be calculated using Bayes' theorem, which is a way to find a probability when we know certain other probabilities.
The problem gives us the following probabilities:
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The probability that a person is guilty given that they tested positive, P(Guilty|Positive), is what we're trying to find.
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The probability that a person tested positive given that they are guilty, P(Positive|Guilty), is 1. This is because if a person is guilty, they would definitely test positive.
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The probability that a person is guilty, P(Guilty), is 1/4000. This is because there are 4000 people and only one of them is guilty.
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The probability that a person tested positive given that they are innocent, P(Positive|Innocent), is 1/500. This is the chance of a false positive.
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The probability that a person is innocent, P(Innocent), is 3999/4000. This is because out of 4000 people, 3999 are innocent.
We can plug these values into Bayes' theorem:
P(Guilty|Positive) = [P(Positive|Guilty) * P(Guilty)] / [P(Positive|Guilty) * P(Guilty) + P(Positive|Innocent) * P(Innocent)]
P(Guilty|Positive) = [1 * (1/4000)] / [1 * (1/4000) + (1/500) * (3999/4000)]
P(Guilty|Positive) = 1/4000 / (1/4000 + 3999/2000000)
P(Guilty|Positive) = 1/4000 / (2001/2000000)
P(Guilty|Positive) = 0.111
So, the correct answer is 0.111.
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