Deer ticks can carry both Lyme disease and human granulocytic ehrlichiosis (HGE). In a study of ticks in the Midwest, it was found that 16% carried Lyme disease, 10% had HE, and that 10% of the ticks that had either Lyme disease or HGE carried both diseases. (a) What is the probability that a tick carries both Lyme disease (L) and HE (H) ? (b) What is the conditional probability that a tick has HE given that it has Lyme disease?

a) 0.024

b) 0.148

Explanation:

Let 's represent the set of deer ticks Carrying Lyme disease with L and the set of deer ticks carrying Human Granulocytic Ehrlichiosis with H

Given:

P (L) = 0.16

P (H) = 0.10

P (L n H) = 0.1 ·P (L u H)

Hence, P (L u H) = 10 ·P (L nH)

(a)

Hence. using the equation. P (L U H) = P (L) + P (H) - P (L n H)

Hence, 10 · P (L n H) = 0.16 + 0.1 - P (L n H)

Hence, 11 · P (L n H) = 0.16 + 0.1 = 0.26

Hence, P (L n H) = 0.26/11=0.024

(b)

We know that condition probability P (H ║ L) = p (L n H) / P (L)

hence, P (H ║ L) = (0.26/11) / 0.16 = 0.148