If I roll a die 10 times, what is the true proportion of getting a 6?

The proportion will be the number of sixes you roll divided by the total number of rolls (10).

The expectation is the weighted average of every possible proportion. Manually, this is 10*(1/6)^10 + 9*10(1/6)^9*(5/6) + 8*45(1/6)^8*(5/6)^2 + 7*120(1/6)^7*(5/6)^3 + 6*210(1/6)^6*(5/6)^4 + 5*252(1/6)^5*(5/6)^5 + 4*210(1/6)^4*(5/6)^6 + 3*120(1/6)^3*(5/6)^7 + 2*45(1/6)^2*(5/6)^8 + 10(1/6)*(5/6)^9 + 0(5/6)^10 divided by 10 = 1/6.
Intuitively, it is the number of rolls we are considering (1) divided by the number of possible rolls (6).

Think about it differently. If you roll 10 six sided dice at the same time, you'll have your answer. How many permutations?

If I roll a die 10 times,
what is the true proportion of getting a 6?

The probability of getting at least one 6 is 0.8384944171.