KöMaL 2019 B5018

The sultan imprisoned all the 1024 mathematicians of his empire. They were not allowed to keep any of their possessions except for a single copper coin each. The mathematicians know that there are 1024 of them, but they are not able to communicate with one another in any way. On his birthday the sultan offered them the following game: they are taken to the prison yard one by one. Each of them may say either 0 or 1 when taken there. If the sum of the numbers they say is 1, then he will let them all go free. (The mathematicians cannot signal to each other, they do not know how many others have been to the yard before them, or what those before them have done in the yard.) What are the chances that they can get out of the prison?