Calculating the probability of someone having zero to n diseases given overlapping prevalence I'm trying to calculate the proportion of people in a population who have a combination of disease $A$, $B$ and $C$. This can range from $0,0,0$ (the set of people with no disease) to $1,1,1$ (the set of people with all three diseases).
I would like to calculate the probability each of these 8 outcomes (from $0,0,0$ to $1,1,1$) given three inputs, which are the prevalence of each disease (e.g. 80% of people have disease $A$, 70% of people have disease $B$ and 50% of people have disease $C$). I would also like to generalise this to $n$ diseases in the future. It is important to be able to distinguish between each outcome, because they have different effects on people (e.g. $0,1,1$ has a different chance to kill someone than $1,0,1$).
Given that we know nothing about the relationship between these diseases and that the prevalences do not sum to 100%, I do not think this has a single solution. However, if we expect the diseases to be independent of each other, I assume this is solvable, but I do not know how to do it. If I reduce the diseases to two, this becomes easier. For example:
If the prevalence of Disease $A$ is $80\%$ and the prevalence of Disease $B$ is $70\%$:

*

*$1,1$: The proportion of people with $A \& B$ = $0.8 * 0.7 = 0.56$

*$1,0$: People with $A$ only = $0.8 - 0.56 = 0.24$

*$0,1$: People with $B$ only = $0.7 - 0.56 = 0.14$

*$0,0$: People with no disease = $1 - (0.56+0.24+0.14) = 0.06$
But in the case of three diseases, I get stuck. For example:
Prevalence of Disease $A$ is $80\%$, Prevalence of Disease $B$ is $70\%$, Prevalence of Disease $C$ is $50\%$

*

*$1,1,1$ = $0.8 * 0.7 * 0.5 = 0.28$
From here, i'm not sure what to do. Perhaps I need to simultaneously solve the next set of equations e.g. $1,0,1$ and $1,1,0$, but i'm not sure.
Is this on the right track? Or does this problem become unsolvable with $>2$ diseases?

+1 for the query, very good analysis so far.
To carry it into three people, using:

80% of people have disease A, 70% of people have disease B and 50% of people have disease C

and

However, if we expect the diseases to be independent of each other, I assume this is solvable, but I do not know how to do it.

Let $p_a = 0.8 =$ chance of person having disease A. 
Let $p_b = 0.7 =$ chance of person having disease B. 
Let $p_c = 0.5 =$ chance of person having disease C.
Let $q_a = 1 - p_a.$ 
Let $q_b = 1 - p_b.$ 
Let $q_c = 1 - p_c.$
You define 8 possibilities, where each person either does or doesn't have each of the 3 diseases.
Then, each of the 8 possibilities will compute to $s_a \times s_b \times s_c$, where if the possibility refers to:
the person having disease A, set $s_a = p_a,$ else $s_a = q_a.$
the person having disease B, set $s_b = p_b,$ else $s_b = q_b.$
the person having disease C, set $s_c = p_c,$ else $s_c = q_c.$
