The outer shell of the cube of 10 cm encloses a cube of 8 cm.

For the 10 cm cube, Volume =10*10*10= 1000 cm^3
Volume of smaller cube = 1*1*1 = 1 cm^3
Total 1cm^3 in a 10cm cube - 1000/1 = 1000 nos

Volume of 8cm cube = 8*8*8 = 512
Volume of smaller cube = 1*1*1 = 1 cm^3
Total 1cm^3 in a 8cm cube = 512/1 = 512 nos

As excluding smaller 1cm cubes forming the 8cm cube all other cubes will have at least one side painted; ie, = 1000 -512 = 488

So probability = 488/1000 ~~1:2[/b]