What an Odd Function
Jelly, 7 bytes

Æf^&lt;¥4P


Swaps 2s and 3s in the input's prime factorization. The probability of odds is 2/3.

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How it works

Æf^&lt;¥4P  Main link. Argument: n

Æf       Compute all prime factors of n, with duplicates.
    ¥4   Combine the two links to the left into a dyadic chain and call it with
         right argument 4.
   &lt;       Compare each prime factor with 4. Yields 1 for 2 and 3, 0 otherwise.
  ^        Bitwise XOR the results with the corresponding prime factors.
         This maps 2 to 3, 3 to 2, and all other primes to themselves.
      P  Take the product of the resulting primes.