Is it more likely that N contains the digit 1 or that it does not?

I went from 1 to 1,000,000 just to make the mental math a little easier. 

Of these numbers, 1/10 of them (100,000) will have a 1 in the hundred-thousands place. 

Of the (1,000,000 - 100,000) = 900,000 that don’t, 1/10 of them (90,000) will have a 1 in the ten-thousands place.

Of the (900,000 - 90,000) = 810,000 that don’t, 1/10 of them (81,000) will have a 1 in the thousands place.

Of the (810,000 - 81,000) = 729,000 that don’t, 1/10 of them (72,900) will have a 1 in the hundreds place.

Of the (729,000 - 72,900) = 656,100 that don’t, 1/10 of them (65,610) will have a 1 in the tens place.

Of the (656,100 - 65,610) = 590,490 that don’t, 1/10 of them (59,049) will have a 1 in the units place. 

So 100,000 + 90,000 + 81,000 + 72,900 + 65,610 + 59,049 = 468,559 of the numbers contain a 1, and 999,999 - 468,559 = 531,440 do not.  Therefore it is more likely that N does not contain the digit 1.