what does ろう in this sentence mean
息子を保育園に入れるのは不可能だろうと思っていた
I could understand the grammar and the meaning of this sentence but I dont know why there is ろう in ろうと思っていた

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My book says that だろう is less formal version of でしょう and that it means "I suppose" so the speaker is not entirely sure of what he's saying he's thinking.

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If there is not ろう, I think you can say I suppose that it's not possible to ~. But, If there is ろう, you can say I guess that it's possible to~.

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「息子を保育園に入れるのは不可能だと思っていた」
is also OK. The difference of both is not so huge. 
I think you already know this difference, don't you?

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