The TRT would not get any significant snow in October.
Now, we need to define "significant."
I have been snowed upon in every calendar month, somewhere in California. In September and October, there might be "early fall flurries" that don't amount to much. Maybe an inch below 10,000' and maybe a little more above 10,000'. Tahoe does not have much really high elevation. By the end of October, the flurries might last longer.
A few hikers could get all fouled up by an inch of snow. Most would not.