Tukukkk

I need to recover some old data which were stored on a drive. The data were deleted and the hard drive was formatted twice and filled twice with random data intentionally.

First the drive had a Windows 7 installed.

Then it was formatted twice with windows installer. After each format the space was filled with random video files.

Considering that when we delete something from a drive, only the index is deleted, the actual data is removed when new information is overwritten in that section of the drive.

The drive is an SSD.

Is there any possibility to find the old data?

Given the information that you have rewritten all ssd contents twice with true random data to the brim I'd say

No, you cannot recover any data from that disk.

This is the sane answer to give to people who lost data and e. g. show up in a data recovery shop.

If you want an academic answer weather or not it's possible at all then we're entering the hypothetical sphere of "given unlimited money and will - is it then possible?".
There are a lot of contributing factors (e. g. SSD Controller, state of dead cells, random data source, partition alignment, …).
But since you asked folks on the internet instead of physically shredding that disk I assume the disk holds no valuable information for any world power.

Please notice that the "Loose all data" option at your operating system installer does not perform a complete wipe of the disk.

Is Data Remanence a Myth?

This is great coverage of the underlying question -- is data recoverable after a wipe. And while the preponderance of answers agree to be "no", the source documentation does also asert this but acquieses that bits of information are potentially recoverable.

So the answer to your question is "no", you cannot recover whole video files after a byte-by-byte overwrite. However, if the drive is known to contain text based data of significant interest where fragments may be enough to piece together a provactive picture, the answer becomes less definite. But, in those cases, you'd be talking in the realm of corporate espionage by the biggest companies in the world and/or nation-states.

You said the hard drive is SSD. This is enough information to answer with a resounding no - even with a single pass write. Let me explain why ...

With an SSD when you delete a file the operating system sends a TRIM command to the SSD and the SSD will delete said file completely. This happens immediately. Why? Because it's faster for the SSD/OS to work this way.

You can read more about it here. Provided the OS issues the TRIM command and the SSD acts upon it then a 1 pass write is enough.

Any more than that and you're just going to burn your SSD out quicker.

It depends on what you mean when you say “filled with random video files”.

If the drive was completely filled to capacity with new data, it is highly unlikely to be retrievable.

If the drive was filled to 50% of capacity with new data, the chances are better, but not great.