First coin toss still ahead after ten tosses

A fair coin is tossed $10$ times. Call a face ahead after $n$ tosses if it has come up in strictly more than half of those $n$ tosses. What is the probability that whichever face appeared on the very first toss is ahead once all $10$ tosses are done?

$\tfrac12$. Say the first toss is heads. Heads is ahead after $10$ tosses exactly when the remaining $9$ tosses contain at least $5$ heads. With an odd number ($9$) of fair tosses, at least $5$ heads and at most $4$ heads are mirror images of each other, so each has probability $\tfrac12$.