Six dice are thrown simultaneously. The probability that all of them show the same face, is

$\frac{1}{6^5}$

The total number of elementary events associated to the random experiment of throwing six dice is 

$6 × 6 × 6 ×6×6× 6= 6^6$

All dice show the same face means we are getting same number on all six dice, this number can be any one of the six number 1, 2, ...., 6. The number of ways of selecting a number is ${^6C}_1$.

∴ Favourable number of elementary events = ${^6C}_1$.

Hence, required probability $=\frac{^6C_1}{6^6}=\frac{1}{6^5}$