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 HGE, and that 10%
of the ticks that had either Lyme disease or HGE
carried both diseases.
(a) What is the probability PLH that a tick carries
both Lyme disease (L) and HGE (H)?
(b) What is the conditional probability that a tick
has HGE given that it has Lyme disease?

It comes from:
\( P(L\cap H)=0.1P(L\cup H) \)
or
\( 10P(L\cap H)=(L\cup H) \)
So
\( 10 P(L∩H)=P(L)+P(H)−P(L∩H) \)
If we transpose \( -P(L\cap H) \) to the left, we get
\( 11P(L∩H)=P(L)+P(H) \)
and the rest follows.