Abstract: This paper derived the probability formula for a chordal $SLE_{\kappa}$ trace across a given point in upper half plane $\mathbb{H}$. And on the basis of the left-passage probability, established the probability formula for a critical percolation in a closed unit circle without a given point. Finally, according to the relationship of exploration process and $SLE_{6}$, got that with the same starting and ending point, the trace of $SLE_{6}$ has the same distribution of self-avoiding walk.

Key words: chrodal $SLE_{\kappa}$, Loewner equation, harmonic measure, critical percolation