With an All India Rank (AIR) of 39,650 and belonging to the 2B category (OBC) in Karnataka, you have a reasonable chance of securing a government seat in an Anaesthesia postgraduate program through either state counseling or the All India Quota (AIQ). However, your chances depend on a few key factors:

1. **Previous Year Cutoffs**: Reviewing the previous year’s cutoffs for Anaesthesia in Karnataka government colleges can give you a good idea of your chances. In Karnataka, for PG medical courses, the cutoffs tend to vary based on the college, specialty, and category. Typically, the cutoffs for the 2B category are slightly lower than the general category, improving your chances.

2. **State Counseling**: In Karnataka, state counseling offers a significant number of PG medical seats under the state quota. Since you belong to the 2B category, you may have an advantage due to the reservation policies. Check the state quota cutoffs for Anaesthesia from previous years to assess your position.

3. **All India Quota (AIQ)**: Under the 15% AIQ, the competition tends to be tougher. While Anaesthesia is a less competitive branch compared to some others like Radiology or Dermatology, you may still have a chance depending on seat availability and the cutoff trends for your category.

Keep track of the official counseling notifications and cutoffs to understand your admission prospects better.