Question

Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a recent study, suppose that of all ticks in a certain location carry Lyme disease, carry HGE, and 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?

Answer 1

The probability that a tick carrying HGE is also a carrier of Lyme disease is 1.

Step-by-step explanation:

To find the probability that a randomly selected tick that carries HGE is also a carrier of Lyme disease, we need to use conditional probability. Let P(A) represent the probability of a tick carrying HGE, and P(B) represent the probability of a tick carrying Lyme disease. We are given that P(B|A) = 0.5, which means the probability of a tick carrying both HGE and Lyme disease is 0.5. Using this information, we can use Bayes' theorem to find the probability of a tick carrying Lyme disease given that it carries HGE:

P(B|A) = P(A|B) * P(B) / P(A)

P(B|A) = 0.5 * P(B) / P(A)

We can rearrange this equation to find P(B):

P(B) = P(B|A) * P(A) / 0.5

Since we are given that the ticks that carry at least one of the diseases carry both of them, we have:

P(A) = P(B)

Substituting P(A) for P(B), we get:

P(B) = P(B|A) * P(B) / 0.5

P(B) = 0.5 * P(B) / 0.5

Cancelling out the 0.5 on both sides, we find:

P(B) = P(B)

In other words, the probability that a randomly selected tick that carries HGE is also a carrier of Lyme disease is 1. Both diseases are carried by the same ticks, so if a tick carries HGE, it is guaranteed to also carry Lyme disease.