Well, since it involves dividing one infinite (number of real numbers contain 3 as a digit) by another (number of real numbers), and the infinites are of the same order, the answer is technically undefined.
But, in a less rigorously mathematical way of looking at things, you could say that almost all real numbers contain 3 as at least one of their digits.
As an approximation to show this, we can look at percentages of real numbers that do not have 3 at a particular spot.
In the 1's column, 90% of real numbers do not have a 3.
Of that 90%, 90% do not have a 3 in the 10s column.
Of that 90% of 90%, 90% do not have a 3 in the 100s column.
So for each additional column that you examine (whether in front of the decimal point or after) the pool of real numbers without a 3 in them is reduced to 90% of what it was.
It takes only 8 digits to get below 50%. At 23 digits, you're under 10%. At 45 digits, you're under 1%. And as the number of digits goes out to infinity, the percentage of numbers that do not have a 3 in them becomes ever smaller.