Dear ticks can be carriers of either Lyme disease or HGE. Based on a recent stud

Question

Dear ticks can be carriers of either Lyme disease or HGE. Based on a recent study, suppose that 16% of all ticks in a certain location carry Lyme disease, 10% carry HGE, and 10% of the ticks that carry at least one of these diseases in fact carry both of them. If a randomly selected tick is found to have carried HGE, what is the probability that the selected tick is also a carrier of Lyme disease?

Explanation / Answer

We are looking for P(Lyme disease | HGE). P(Lyme disease) = .16, P(HGE) = .1, and P(HGE AND Lyme disease | at least one) = .1. To find the answer, we need P(Lyme disease AND HGE). P(At least one) = P(Lyme disease OR HGE) = P(Lyme disease) + P(HGE) - P(Lyme disease AND HGE) = .16 + 1. - P(HGE AND Lyme disease | at least one)* P(at least one) = .26 - .1*P(at least one) => 1.1*P(at least one) = .26 => P(at least one) = .26/1.1 = .236. P(HGE AND Lyme disease) = P(HGE AND Lyme disease | at least one) * P(at least one) = .1*.236 = .0236. So, P(Lyme disease | HGE) = P(Lyme disease AND HGE)/P(HGE) = .0236/.1 = .236. Hope that helps!

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