we know that
It is much easier to compute the number of numbers that do not contain a [tex] 3 [/tex].
In the first [tex] 10 [/tex] numbers [tex] 9 [/tex] do not contain a [tex] 3 [/tex].
In the first [tex] 100 [/tex] numbers [tex] 81 [/tex] do not contain a [tex] 3 [/tex]. (This is because these numbers are formed by selecting a digit out of
[tex] [0, 1, 2, 4, 5, 6, 7, 8, 9] [/tex] for the first position and another digit out of the same set for the last position and this may be done in
[tex] 9*9 = 81 ways. [/tex]
In the first [tex] 1000 [/tex] numbers, reasoning as above, there are
[tex] 9^{3} [/tex] numbers that do not contain a [tex] 3 [/tex].
Now the pattern should be becoming obvious.
In the first [tex] 10^{n} [/tex] numbers there are [tex] 9^{n} [/tex] numbers that do not contain a [tex] 3 [/tex].
So
In the first [tex] 10^{4} [/tex] numbers there are [tex] 9^{4} [/tex] numbers that do not contain a [tex] 3 [/tex].
[tex] 9^{4}= 6,561 [/tex]
[tex] 10,000- 6,561= 3,439[/tex]
Thus
the ratio of those numbers that do contain a 3 over the total is equal to
[tex] Ratio=\frac{3,439}{10,000} \\ \\ Ratio=0.3439 [/tex]
therefore
the answer is
[tex] 0.3439 [/tex]