= Solution
A second opportunity to reject raises the trial's overall <Type I error> above the nominal 10% in general. If $A$ and $B$ denote rejection at the first and second analyses, the overall error is
$$
\mathbb P_0(A)+\mathbb P_0(A^c\cap B).
$$
The cumulative-data <test statistics> are correlated, so neither 20% nor $1-0.9^2=19\%$ is the general answer. The latter requires independent tests, which is not the situation here.
\b[The rejection thresholds and the stopping/continuation rule must be calibrated jointly.] A prespecified <group sequential design> or <alpha-spending function> can allocate a total error budget of 10% across the analyses, using their joint null distribution. A conservative alternative planned in advance is two tests each at 5%, controlled by the <Bonferroni inequality>. An unplanned change to a design with a remaining error budget can be justified by the <conditional error principle>. Here, however, an original single-look test at 10% has no remaining conditional rejection probability after its final nonrejection. Keeping its entire original rejection region and adding a new positive-probability rejection opportunity cannot in general retain a 10% overall error rate. The two-look rule therefore needs prospective joint calibration; an already observed result cannot retrospectively create an unused error budget.
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