I recently bought a second-hand bike from a private seller. On the second day after I bought it, I discovered a problem with the rear tyre, which was not disclosed to me by the seller. The seller let me inspect the bike on the day of deal, but I didn't notice the problem. Was this my fault that I didn't discover the problem before making a deal? Can this be considered dishonesty on the part of the seller? Is there anything I could do to avoid this from happening in my future purchases?
In general, a private seller of second hand goods makes no warranty as to their merchantability or fitness for purpose: the onus is the buyer to find any faults. However, if the seller has taken active steps to conceal the fault for the purposes of the sale then they are responsible. For example the courts have held that a boat placed deliberately on the sand to conceal a hole made the seller responsible while a painting covering a hole in the wall of a house did not because it had been there for many years.
Different laws apply if the seller is in the business of selling second hand goods.
This is a caveat emptor situation and, on the facts as given by the OP, the (private) seller is not responsible to the buyer. The situation might be different if the vendor had been a retail motorcycle dealer.
Caveat Emptor: Latin (noun). Let the buyer be beware. Law: the principle that the seller of a product cannot be held responsible for its quality unless it is guaranteed in a warranty. (Dictionary.com)
It would depend a lot on what the seller did or did not say. If the sale was "as is" which would be the default rule in a sale of used goods in most cases, then you are stuck. If the seller made a promise that it was in good condition, or represented that the bike was in good working condition and had no defects while knowing otherwise, then there might be liability.
Realistically, the money at stake would be too low to make the matter with litigating even if there was a chance that liability could be shown.