Bayesian probability with repeated trials
Solution 1:
This image is from my previous post Why is the disease frequency often included in examples on medical tests' reliability?. Please use its tree diagram and lowermost line as a reference for the following answer:
Given that two tests are independently negative, the probability that Jemima has COVID is $$\frac{0.45\times0.1\times0.1}{0.45\times0.1\times0.1+0.55\times0.86\times0.86}=1.0941\%.$$ Thus, the required odds is $\displaystyle\frac{1.0941}{100-1.0941}=0.01106$ $$=11:1000.$$