Strategy

If it shows tail, switch coin, until you get a head, then keep using this coin forever.
The idea to find the prob*****ility is to consider different cases:
1. the first toss is a head (the probability os which is 1/4)
2. getting the first head on second toss (prob = (3/4)*(1/4))
3. getting the first head on third too (prob = (3/4)*(3/4)*(1/4))
and so on,...

The above cases will give you different positions on ladder when you are sure you have the fair coin, from this moment you start a random walk up and down with the same probability.