<s>[INST] This problem is originally from the 1983 PUTNAM EXAMINATION
In a game, 100 cards numbered from 1 to 100 are shuffled and distributed among 100 players so that each player gets one card. At the same time 50 special cards, numbered from 1 to 50, are also distributed among the same 100 players so that each player gets at most one special card. (It is possible that a given player may receive no special card.)
Let \(S\) denote the random variable that represents the total number of special cards that have the same number as the numbered cards distributed to the players. Define the probability space.
a) Define a probability space for this experiment.
b) What is the value of \(P(\{S=k\})\)?[/INST]