How to correct father's surname during probation period?

Already you have joined the job. Did you inform the employer about the mistake while joining the job?
The best way is to make an Affidavit before a first-class magistrate. It is better to take the help of an advocate so that he will make the affidavit suitably so that you will not face any problems afterwards.
After getting the affidavit you have to apply for a publication of the same in the Gazette. The procedure for the same will also be known to the advocate. There will be an application form. You have to publish a notice in two newspapers. One in the local language in a regional paper and another one in an English paper in a national newspaper. After that, you have to fill out the application form. pay the required fee and attach the copies of the newspaper cutting and the same is to be sent to the concerned authority.

No sir. I haven't informed the employer (department) about the mistake while joining the job.

In the initial stage itself the office documents especially in Government services, should get corrected. Though it is the duty of Personnel or HR or Establishment sections to verify the correctness, the duty lies on the employee or applicant to furnish correct information in the initial stage itself.
It is essential to anybody to get corrected their basic records with similar name spelling, date of birth (In many aadhaar cards I am seeing they have only year instead of date/month/year), voter Id having different spelling in names). It become our responsibility to see the similarity immediately and steps to be taken for correction. Immediately we can correct our aadhaar card with the correct name, date of birth and father's name. With that corrected documents, all other documents should get corrected. As for as SR - Service Record concerned we should get corrected in all sides without waiting for retirement date as it very well affect our retirement benefits.
For everything we should not step up through correspondence but should initiate by direct approach to the concerned departments with necessary documents in original and Xerox copies. While going the sufficient number of copies of the correct documents as some department officials may need for more copies.

You are currently doing your job even though a mistake in your father's surname occurred and your employer did notice the same at the time verification of your records. The best way is to accept your mistake before the employer in writing to your personal department assuring them of your early rectification of the said mistake.
You may meet some experienced lawyer in your area explaining the mistake of your father's name. He will help you in the process of affidavit to be countersigned by the first class magistrate. Once the affidavit is ready showing his correct surname, this has to sent for the advertisement in two Newspapers- one in local Newspaper and the other one in English Newspaper for the wide publicity of the same.
Preserve this cutting and getting the same Xeroxed, send to the personnel department as a proof that corrections have been made.

At present, you are doing the job and found that the father's surname is a mistake in the document. So, you must inform your employer regarding this so that problems may not arise in future. And you should consult a good lawyer for making an affidavit. Once the process of making the affidavit is completed and shows the correct surname then this has to be sent for advertisement in two Newspapers- one in a Newspaper and the other in English Newspapers for national publicity. Next to it, you need o to fill out the application form with the required fee and attach the copies of the newspaper cutting and all are to be sent to the personal department.