To find the probability that Nur chose playground Y given that she chose a climbing frame, we use Bayes' theorem:

\(P(Y | C) = \frac{P(Y \cap C)}{P(C)}\)

First, calculate \(P(Y \cap C)\):

Nur chooses playground Y with probability \(\frac{1}{4}\) and then chooses a climbing frame with probability \(\frac{1}{12}\) (since there is 1 climbing frame out of 12 total pieces of equipment in playground Y).

\(P(Y \cap C) = \frac{1}{4} \times \frac{1}{12} = \frac{1}{48}\)

Next, calculate \(P(C)\), the total probability of choosing a climbing frame:

\(P(C) = P(X \cap C) + P(Y \cap C) + P(Z \cap C)\)

\(P(X \cap C) = 0\) since there are no climbing frames in playground X.

\(P(Y \cap C) = \frac{1}{48}\) as calculated above.

\(P(Z \cap C) = \frac{1}{2} \times \frac{4}{16} = \frac{1}{8}\) since there are 4 climbing frames out of 16 total pieces of equipment in playground Z.

\(P(C) = 0 + \frac{1}{48} + \frac{1}{8} = \frac{1}{48} + \frac{6}{48} = \frac{7}{48}\)

Finally, substitute back into Bayes' theorem:

\(P(Y | C) = \frac{\frac{1}{48}}{\frac{7}{48}} = \frac{1}{7}\)