First consider all odd numbers: select all odd numbers and arrange them
=>20P10 ways.
Now consider 2: Fix 2 in first place available after arranging all odd numbers
=> 1 way.
Now consider even numbers other than 2: arrange all 9 of those even numbers
=> 9! ways
Total ways of arranging numbers when 2 at an earlier position than any other even number: 20P2 * 1 * 9! ways
Total ways of arranging 20 numbers: 20! ways
=> Probability = $\frac{20P10 * 1 * 9!}{20!}$
= $\frac{20! * 9!}{10! * 20!}$
= $\frac{1}{10}$