= Solution
Under the intended additional assumption of mutually <independent>, continuously calibrated null <p-values>, each test fails to reject with <probability> $0.95$. The <familywise error rate> is therefore
$$
\boxed{\mathbb P(\text{at least one false rejection})=1-0.95^{10}\simeq0.4013.}
$$
There is a qualification to the printed use of “not correlated”: zero pairwise <correlation coefficients> do not generally imply mutual <independence>. A <multivariate normal distribution> with zero off-diagonal <covariances> does give <independence>, but an arbitrary biomarker distribution need not. The formula above requires <independent> rejection events, not just uncorrelated measurements.
The usual intuition for positively dependent tests is that they reject together, reducing the effective number of opportunities for a false positive. A rigorous sufficient assumption is <positive association of random variables> for the rejection indicators $I_1,\ldots,I_{10}$. Products of their non-rejection indicators are decreasing functions; association and induction give
$$
\mathbb P(I_1=\cdots=I_{10}=0)\geq\prod_i\mathbb P(I_i=0)=0.95^{10}.
$$
Thus \b[under this stronger positive-dependence assumption the <probability> is no larger], and perfect dependence gives $0.05$ instead of $0.4013$. Strict inequality is not forced by every form of positive dependence.
Pairwise positive <correlation coefficients> alone, as printed, is insufficient. For a counterexample choose a vector of ten <exchangeable random variables> that are rejection indicators as follows: with <probabilities> $0.545,0.450,0.005$, respectively, reject none, reject one uniformly selected test, or reject all ten. Then
$$
\mathbb P(I_i=1)=0.05,\qquad \mathbb P(I_i=I_j=1)=0.005>0.05^2\quad(i\ne j),
$$
but the <familywise error rate> is $0.455>1-0.95^{10}$. This can be realized with valid null <p-values>: conditional on the indicators, independently draw $p_i$ uniformly on $(0,0.05)$ if $I_i=1$ and on $(0.05,1)$ otherwise. Every $p_i$ is uniform on $(0,1)$, and $\operatorname{Cov}(p_i,p_j)=0.25\operatorname{Cov}(I_i,I_j)>0$. Taking biomarker statistics $X_i=1-p_i$ gives pairwise positively correlated null statistics with exactly these one-sided tests. Therefore the unconditional larger-or-smaller claim needs a specified dependence model.
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