Suppose we have compare the value of a continuous variable $x$ between two groups, A and B, of $n$ samples each. The true distributions of $x$ in $A$ and $B$ are $N(\mu\_A,1)$ and $N(\mu\_B,1)$ respectively, with $\mu\_B>\mu\_A$. We perform a two-sample test with confidence level $\alpha$ using the null hypothesis $H\_0$ that $\mu\_A=\mu\_B$.

My friend asked me to find the probability that we fail to reject $H\_0$. Wouldn't this just be $\alpha$?