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Recognitions:      Homework  Help  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 $$\Omega$$ would be the set of all $$2^{100}$$ sequences, from all Ts through all Hs.  One particular $$\omega$$ would be this one:  $$\omega = \underbrace{HH \cdots H}_{\text{length 50}} \overbrace{TT \cdots T}^{\text{length50}}$$  for the r.vs I defined, and for this $$\omega$$,   $$X(\omega) = Y(\omega) = 50$$   Notice the incredible amount of savings we have in the move from the original sample space $$\Omega$$, which has $$2^{100}$$ elements, to the set of values of $$X$$ (and of course $$Y$$) - there are only 101 different values to "keep track of".  I hope this helps.
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