A fair coin is flipped nine times. Which is more likely, having exactly four heads or having exactly five heads?
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 $\frac{\left(\begin{array}{l}9 \\ 4\end{array}\right)}{2^{9}}=\frac{\left(\begin{array}{l}9 \\ 5\end{array}\right)}{2^{9}}=\frac{126}{512}$.