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Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards he has taken and calls that his score. Let

p

be the probability that his score is a multiple of 13. There exists relatively prime positive integers

a

and

b,

with

b

as small as possible, such that

|p - a/b| < 10^{-10}.

What is

a/b?

Proposed by Dilhan Salgado, Daniel Li

Problem Statement