If you want to solve for the probability that any cards from card N to card (N-M) are in the left deck, I guess you could try: That looks too complicated for me to attempt without a more Ah-ha approach.
I can calculate the probability that at least one of card N-1 or card N-2 falls in the left pile. I think this might be extendable to larger numbers, albeit with the equations becoming somewhat unwieldy.
This somewhat reminds me of the story of the surprise execution.
A prisoner was sentenced to death with the execution to take place within one week. But the judge also stipulated that the prisoner's execution should come unexpectedly, as it would be cruel to have him know that death would be imminent within 24 hours. The prisoner knew all of this.
The prisoner reasoned that he could not be executed seven days after the sentence, because if he were not executed on the sixth day then he would know that he would be executed on the seventh, and that would violate the judge's ruling. For all practical purposes, the prisoner would have to be executed within six days. But that means that he couldn't be executed on the sixth day – if he made it past the fifth day and he knew he couldn't be executed on the seventh day, then he would know that he was to be executed on the sixth day, which would violate the judge's ruling. Continuing on with that sort of logic, the prisoner concluded that he could not be executed at all!
The prisoner was then executed two days after the sentence, which came as a complete surprise to him in accord with the judge's ruling.