= Solution
Daisy's random choice of X or Z agrees with Alice's independent random basis with <probability> $1/2$. Agreement gives the correct label with certainty and disagreement gives it with <probability> $1/2$. Hence
$$
\boxed{P(\text{correct bit}\mid\text{random X/Z})=\frac34,}
$$
the same result as a fixed Z <quantum measurement>[measurement]. Randomizing her basis does not supply the missing information about Alice's choice. These are the <probabilities> for the two specified strategies, rather than an assertion that they are optimal among all possible quantum <quantum measurements>[measurements].
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