Let A be the event "Double headed coin selected" and B be the event "Four heads thrown" ...

Then the formula
P(A|B) = P(B|A) P(A)/P(B)
reduces to
P(A|B) = P(A)/P(B) {1}
Since
P(A) = 1/3
and
P(B) = P(B|A) P(A) + P(B|not A) P(not A) = 1 * 1/3 + 1/16 * 2/3 = 18/48
Substituting into {1} gives
P(A|B) (1/3)/(18/48) = 16/18