The question itself is very short and sweet, and requires no background.
Find a solution to the following system: $$ \left\{\begin{array}{l} a,b,c,d,e,f>0\\\\ a+b+c+d+e+f=1 \\\\\displaystyle\frac{c}{a+c}\geqslant .999 \\\\\displaystyle\frac{d}{b+d}\leqslant .001 \\\\\displaystyle\frac{c}{c+d+e}\leqslant .001 \\\\\displaystyle\frac{d}{c+d+e}\geqslant .999 \end{array}\right. $$ I don't need to find the entire solution set. I literally just need one particular solution. It's basic algebra but it's giving me a headache so maybe someone has some software they can use to find it.
Why am I asking?
Philosopher Robin Collins wrote a paper where he claims that if $H$ and $K$ are competing hypotheses and $E$ is observed evidence with $P(E|H)\gg P(E|K)$, then $E$ is "strong evidence" for $H$ over $K$. This is an ambiguous claim, but my guess is that he means something like the following:
(LP) If $H\cap K=\emptyset$ and $P(E|H)\gg P(E|K)$, then $P(H|E)\not\ll P(K|E)$.
I believe that (LP) is false, but I'm struggling to produce a counterexample. In particular, I need a counterexample where $P(H\cap K)\neq 1$, $.99\leqslant P(E|H),P(K|E)<1$, and $0<P(E|K),P(H|E)\leqslant .01$. I then assign variables based on the following Venn diagram:
Finding a solution to the system will disprove (LP).
Let's rewrite things in a more convenient form. $${c\over a+c}\ge.999{\rm\ is\ }c\ge999a$$ $${d\over b+d}\le.001{\rm\ is\ }b\ge999d$$ $${c\over c+d+e}\le.001{\rm\ is\ }{d\over c}+{e\over c}\ge999$$ which will be satisfied if $d\ge999c$, $${d\over c+d+e}\ge.999{\rm\ is\ }{c\over d}+{e\over d}\le{1\over999}$$ which will be satisfied if $d\ge2000c$ and $d\ge2000e$. So let's take $b=.5$, $d=.0005$, $c=e=.0000002$, $a=.0000000002$, and $f=1-a-b-c-d-e$.