= Solution
Let $D$, $C$, and $V_0$ denote the three disjoint categories, and $+$ or $-$ the test result. The <law of total probability> gives
$$
\boxed{\mathbb P(+)=0.1(0.95)+0.2(0.9)+0.7(0.05)=0.31.}
$$
To calculate the requested <conditional probability>, first combine the two categories with the virus. Their joint negative-test <probability> is $0.1(0.05)+0.2(0.1)=0.025$, whereas $\mathbb P(-)=1-0.31=0.69$. <Bayes' theorem> therefore gives
$$
\boxed{\mathbb P(D\cup C\mid-)=\frac{0.025}{0.69}=\frac5{138}\approx0.03623.}
$$
These numbers use the specified population proportions and test <probabilities>; the negative result reduces the initial combined proportion $0.3$ to about $3.62\%$ within this model.
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