Learn Turkish: Zavallı köpeğin kulübede olma ihtimali yok. - There is no chance the poor dog is in the kennel.

It acts as the owner of the 'possibility of being' (olma ihtimali). In Turkish, to express 'the chance of the dog being,' you build a structure literally meaning 'the dog's possibility of being.' The genitive -in connects the dog to the compound noun phrase that follows.

How does the grammar of olma ihtimali work?

Olma is a verbal noun (often called a gerund) meaning 'being,' formed by adding -ma to the root ol-. When paired with ihtimal (possibility), it creates a compound noun: olma ihtimali (possibility of being). The -i at the end is the possessive suffix that groups the two words together.

I've learned that verbal nouns with a specific subject usually take a possessive suffix, like köpeğin olması. Why is it just olma here?

That is a great observation. While you could technically say köpeğin kulübede olması ihtimali (the possibility of the dog's being in the kennel), Turkish speakers frequently treat olma ihtimali as an inseparable set compound. Because of this, the structure simplifies to 'the dog's [being-possibility],' leaving olma bare without its own possessive suffix.

Could I use the word fakir instead of zavallı for 'poor'?

Not in this context. Fakir (or yoksul) means 'poor' in the financial sense of lacking money. Zavallı means 'poor' in the sense of being pitiable, unfortunate, or helpless, which is the only type of 'poor' that makes sense for an animal in a sad situation.