Three Heads Before Three Tails

Look for a symmetry between heads and tails.

Swap every H and T in any complete path.

1. 1. Pair the outcomes

   Every sequence ending first in HHH pairs uniquely with the sequence obtained by swapping H and T, which ends first in TTT.

2. 2. Apply symmetry

   The paired events have equal probability and one must occur.

   $$P(HHH\text{ first})=P(TTT\text{ first})=\frac{1}{2}$$

Answer: 1/2