Let $W_t$ be a standard Wiener process and $\tau = \min \lbrace t \geq 0 :W_t \geq a \rbrace$, the first time the process reaches level $a$. By symmetry of the Gaussian distribution we have $$ P(W_t - W_\tau > 0 \mid \tau =s) = P(W_t - W_\tau < 0 \mid \tau =s) = 1/2 $$ Now the statement I don't fully follow is:
Integration over $0 \leq s \leq t$ against the distribution of $\tau$ gives $$ P(W_t - W_\tau > 0 \text{ and } \tau <t) = \frac{1}{2}P(\tau < t) $$
This is taken from these notes (page 7): http://galton.uchicago.edu/~lalley/Courses/390/Lecture5.pdf
I am unsure of what exactly one integrates against the distribution of $\tau$, and whether this distribution is used explicitly or not, and which rule of probability one uses to assert that integration against the distribution is the step to perform.
If someone could show the intermediate steps it would be greatly appreciated.
\begin{align} P(W_t - W_\tau > 0 \text{ and } \tau <t)&=P(W_t - W_\tau > 0 \text{ and } \tau <t\Big|\tau<t)P(\tau<t)\\&+P(W_t - W_\tau > 0 \text{ and } \tau <t\Big|\tau>t)P(\tau>t)\\ &=P(W_{t-\tau} > 0 )P(\tau<t)+0\times P(\tau>t)\\ &=\frac12P(\tau<t) \end{align}