Here are my thoughts: $P (W_{t} < a | W_{2t} > a) = \frac{P(W_{2t} > 2a, W_{t} < a)}{P(W_{2t} > 2a)} = \frac{P(W_{2t} - W_{t} > a,\, W_{t} < a)}{P(W_{2t} > 2a)}$
Then, taking into account the independency of increments of a Wiener process, this equals to: $$ \frac{P(W_{2t} - W_{t} > a) * P(W_{t} < a)}{P(W_{2t} > 2a)} $$ And then, all the components of this expression are known. I was wondering if I made some mistakes in the steps above. Or my solution is legit?