KöMaL 2020 C1636

The Hungarian poet Dezső Kosztolányi spent a few weeks in Paris when he was a student. When he was given for change a ten-centime coin not in circulation any more, he wanted to give it away. He did not succeed, which he explained to himself by the expression on his face revealing his intentions. Therefore he decided to get 9 valid ten-centime coins, mix them with the worthless coin in his pocket, and by not looking at them he pays with one of them in a shop. He continued doing so until he had a single coin in his pocket: the coin out of circulation. What is the probability of this?