= Solution
Let $p=|\alpha|^2$ and $1-p=|\beta|^2$. The first <basic measurement> has outcome <probabilities> $p$ and $1-p$, obtained by conjugating the amplitudes in the bras. Conditional on its first outcome, the next <computational-basis measurement> returns 0 with <probability> $p$; conditional on its second outcome, that <probability> is $1-p$. The <law of total probability> therefore gives
$$
\Pr(\text{second outcome }0)=p^2+(1-p)^2=\frac12+2\left(p-\frac12\right)^2.
$$
Consequently the \b[minimum is $1/2$, attained exactly when $|\alpha|=|\beta|=1/\sqrt2$]. The relative complex phases do not affect this result. The alternatives after the first measurement form a classical mixture; their amplitudes must not be added coherently as if no measurement occurred.
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