From first principles, an outcome to any sequence of 100 flips consists of a sequence of H (for Heads) and T (for Tails) of length 100, so [tex] \Omega [/tex] would be the set of all [tex] 2^{100} [/tex] sequences, from all Ts through all Hs.
One particular [tex] \omega [/tex] would be this one:
[tex]
\omega = \underbrace{HH \cdots H}_{\text{length 50}} \overbrace{TT \cdots T}^{\text{length50}}
[/tex]
for the r.vs I defined, and for this [tex] \omega [/tex],
[tex]
X(\omega) = Y(\omega) = 50
[/tex]
Notice the incredible amount of savings we have in the move from the original sample space [tex] \Omega [/tex], which has [tex] 2^{100} [/tex] elements, to the set of values of [tex] X [/tex] (and of course [tex] Y [/tex])  there are only 101 different values to "keep track of".
I hope this helps.
