The story

Hercules just got another task. This time he has to steal Geryon’s golden cow from the far-off Mediterranean island of Erytheia. Hercules knows that Geryon has a large flock of cows among which there is only one golden cow, which Hercules needs to retrieve.

Hercules is tired of fighting and challenges Geryon to a bloodless battle instead. The game Hercules suggests is to be played with the remaining cows.

The game

Each player chooses to eliminate 13 or 50 or 63 cows from the flock at each turn. The player who eliminates the last set of cows gets the golden cow.

Geryon is not very good at games, so he does not tell Hercules how many cows are there in his flock, lest he thinks of a winning strategy, however in his foolishness he lets slip that the number lies anywhere between 0 to 205530. He also forces Hercules to decide who shall begin the game.

Hercules now wants to know what the odds are of him losing the game if he goes first and plays perfectly, given that the number of cows lies anywhere between 0 and 205530.

Can you help Hercules by finding the probability? The answer must be correct to 4 decimal places.