Is this rearrangement in carbocation possible? Generally, in the following type of reactions, positive charge on $\alpha$-Carbon causes ring expansion, but in the following example positive charge is formed on $\beta$-Carbon and the cation is cyclo-propyl methyl. 


So is the above rearrangement even practically possible?
yes,  the rearrangement in carbocation is possible depending upon the condition of reaction and the stability of the new carbocation formed
 eg- the general order of stabilty for carbocation is 3degree >2degree > 1degree 
NOTE- the stability of a carbocation depends on aromaticity resonance hyperconjucation inductive effect and sometime ring expansion also occurs