Coin Race #2

You flip a fair coin 10 times and keep a running count of Heads and Tails. After the 10th flip, Heads is "winning the race" if there are strictly more Heads than Tails (a 5-5 tie is not a win for Heads).

Given that the first flip is Tails, what is the probability that Heads is winning the race after all 10 flips? Give the exact value as a fraction or a decimal.