Let A be the event "diagnosed with the disease".
"10 % of the people over 30 years have a certain type of disease": Pr(D) = 0.1 => Pr(D') = 0.9
"The probability of a doctor correctly diagnosing a person with the disease is 78%": Pr(A | D) = 0.78 => Pr(A' | D) = 1 - 0.78 = 0.22
"The probability of a doctor ... incorrectly not having the disease 8%": Pr(A | D') = 0.08 => Pr(A' | D') = 0.92 (well, this is the only interpretation of the wording that makes any sense to me, anyway)
a. Pr(A) = Pr(A | D) Pr(D) + Pr(A | D') Pr(D') = ....
b. You need Pr( D | A).
Note that and .
This is just another way of writing the usual formula .
Now substitute the values of Pr( A | D), Pr(D) and Pr(A).