I don't think the OP is asking about a 51% attack. My understanding of the OP's question is that he is asking about the probability of there being a chain of orphaned blocks that is 6 blocks deep. The OP does use the words "double spend attack", but the context of his question appears to be asking about orphaned blocks.

When a block is found, it will take x amount of time to propagate throughout the network, and if another miner finds a competing block within that time, there is the potential for some of the network to be working on top of one block at height y, and some of the network to be working on top of another block at the time height. If this were to happen, and both parts of the network find block y + 1 within the aforementioned x time, there will be a chain of two orphaned blocks.

Without knowing how long it typically takes for blocks to propagate, it is difficult to calculate the chances of a chain split that is n blocks deep. I do know that the miners, and mining pools have invested heavily in getting their found blocks to the other miners quickly, and over time, the time required for a block to propagate has been reduced.

With the above being said, just because there is a chain of orphaned blocks, it is not necessarily going to be true that there will be a different set of confirmed transactions in each of the blocks. In general, miners will fill their blocks with transactions based on transaction fee rate, so all miners are more or less expected to include the same transaction set in each of their blocks, notwithstanding transactions that were very recently broadcast immediately prior to a block being found.