～のだろうか【のだろうか】

のだろうか is a question you ask yourself: "I wonder", "could it be that". Verbs and い-adjectives take it in the plain form, past or present; a noun or a な-adjective takes な in the present.

• × 幸せだのだろうか → 幸せなのだろうか

With a question word

With なぜ, 何 or 誰, the sentence becomes a question nobody is expected to answer. Essays and stories use it to raise a point, and it often runs through someone's thoughts.

のだろうか or かな

のだろうか is serious and reflective, and belongs in writing and in careful speech. In conversation you'd say かな for the same thought, or んだろう with の shortened to ん.

Asking someone: のでしょうか

のだろうか is aimed at yourself. To put the same question politely to someone else, use のでしょうか.