Can SN1 reactions occur in compounds where the carbocation, even though is is initially $1^\circ$, can rearrange to form a more stable carbocation?
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Your thinking is correct that they can rearrange and then form stable carbocation (though your first example is wrong as it would become anti-aromatic after rearrangement) , but the answer to your question is NO , they would be substituted by SN2 mechanism being 1 degree . The reason behind this is that the compound will not try to form a carbocation first where it is not that stable , you can think of it in this way that at the initial stage the compound doesn't know that it can rearrange later to form stable carbocation, this is something that we can see but not the compound and thus it will go by normal SN2 mechanism.