Both outcomes are equally likely! Notice that for every sequence with four heads (such as HTTHHTHTT) there is a corresponding sequence with exactly five heads formed by reversing all the coin flips (in this case $T H H T H T H H$ ). Thus the number of sequences of nine flips with exactly four heads is equal to the number of sequences with exactly five heads; since every sequence is equally likely this completes the proof.

In fact, both probabilities are equal to $\frac{\left(\begin{array}{l}9 \\ 4\end{array}\right)}{2^{9}}=\frac{\left(\begin{array}{l}9 \\ 5\end{array}\right)}{2^{9}}=\frac{126}{512}$.