Probability of infection by staphylococcus aureus Please forgive my innumeracy, but I have a question with which I am hoping someone might be able to help me.
Suppose the following be true. The chance of a prosthetic hip joint becoming infected by staphylococcus aureus is one per cent. The chance of a natural hip joint becoming infected by staphylococcus aureus is 0.1%. In other words (in case I am misusing the word 'chance'), one in one hundred people with prosthetic hip joints will become infected by staphylococcus aureus, whereas only one in one thousand people with natural hip joints will become so infected.
Now suppose that X has a prosthetic hip joint and that X's hip joint becomes infected by staphylococcus aureus.
Given only the information provided here, is it correct to say that X's hip joint probably would not have become infected but for the fact that X has a prosthetic hip joint (instead of a natural hip joint)? In other words (to make it clear what I mean by 'probably'), is it correct to say that there is a greater than 50 per cent chance that X's hip joint would not have become infected but for the fact that X has a prosthetic hip joint? Why or why not?
 A: It is tempting to think of the cases where infection and prosthetic hip joints occur as a superset of the cases where natural hip joints contract the infection. This way we could reach a result that "Only 10% of prosthetic hip joint infections would have been infected with a natural hip joint" but this is not a valid conclusion. The two groups aren't directly comparable. We'd need some data about the correlation of these percentages, which is impossible to acquire because we can't simulate alternate universes where person X who has a prosthetic hip joint doesn't have the prosthetic hip joint.
Even if we could correlate the percentages of people that get a staphylococcus aureus infection in either group, the statement "X's hip joint probably would not have become infected but for the fact that X has a prosthetic hip joint" implies a causal link. We know that correlation doesn't imply causation.
In fact, given that X was any other person with a prosthetic hip joint, they would probably not have acquired the hip joint infection. The probabilities in this case work out similarly.
A: 
Now suppose that X has a prosthetic hip joint and that X's hip joint becomes infected by staphylococcus aureus.


Given only the information provided here, is it correct to say that X's hip joint probably would not have become infected but for the fact that X has a prosthetic hip joint (instead of a natural hip joint)?

Let $A$ denote the event that hip joint becomes (or became) infected. 
Let $B$ denote the event that hip joint prosthetic. 
Let $C$ denote the event that hip joint natural, so $p(B) + p(C) = 1.$
Although the following analysis is not directly on point, it may be deemed pertinent.
That is, it may be pertinent to use Bayes Theorem, to calculate 
$$p(B|A) = \frac{p(AB)}{p(A)}.\tag1$$
Note that this is not equivalent to the question that the OP is asking.  However,
re the OP's query, $p(B|A)$ may still be considered relevant.
The only data given in the problem is that:
$$\frac{p(AB)}{p(B)} = p(A|B)  = 0.01 \tag2$$ and 
$$\frac{p(AC)}{1 - p(B)} = \frac{p(AC)}{p(C)} = p(A|C) = 0.001. \tag3$$
Since $B$ and $C$ are contradictory events, the roadblocks
to solving equation (1) above are

*

*computing $p(AB)$

*computing $p(A) = p(AB) + p(AC).$
Since $p(B)$ and (therefore) $p(C)$ are unknown, I see no way of 
computing either $p(AB)$ or $p(AC)$.
However, it is reasonable to let $X = p(B)$, see where this leads,
and then make real world guesstimates of $p(B)$.  This will then
provide a real world estimate response to the query.
$p(AB) = 0.01 \times X.$ 
$p(AC) = 0.001 \times (1 - X).$
Therefore, $p(A) = 0.001 + 0.009X.$
Therefore, $p(B|A) = \frac{0.01X}{0.001 + 0.009X}.$
Taking a step back, it is clear that for small values of $X$, the
denominator in the above fraction will be $\approx 0.001$.
Anyway:
As an example, suppose that $X = 0.001.$
Then, $p(B|A) = \frac{.00001}{0.001009} \approx \frac{1}{100}$.
Suppose instead that $X = 0.01.$
Then, $p(B|A) = \frac{.0001}{0.00109} \approx \frac{1}{10}$.
Suppose instead that $X = 0.1.$ (An estimate that is clearly too high).
Then, $p(B|A) = \frac{.001}{0.0019} \approx \frac{10}{19}$.
Therefore, one concludes that $p(B|A)$ is probably not much above 
$\frac{1}{10}$, and is perhaps significantly less.
Intuitively, the idea is that prosthetics only increased the chance of
infection by a factor of $(10)$, while the real world chance that
someone has a prosthetic is perhaps exponentially below $\frac{1}{10}.$

Now the fun starts.
The sample values examined for X suggest that
$p(B|A) \approx 10 \times p(B)$. 
This strikes me as a reasonable guesstimate, since the chance of infection increases
by a factor of 10, given the prosthetic.
At this point, my only evaluation is that the whole issue is unclear.
In other circumstances, I might attempt to belabor this point, but the answer of jMdA
has already said everything that I might have thought of.  Lacking real world data on causation, but understanding the math as I do, my suspicion is that there is
an unknown causal link between prosthetics, and the increased chance of infection.
In fact, I will go further, and say that it is close to a virtual certainty that the
causal link exists, because how else could you explain that the chance of infection increases by a factor of 10, given the prosthetic?
Note that within the narrow confines of the OP's query, it is irrelevant whether the link is cellular, or because people with prosthetics might be more likely to have improper sleeping habits or dietary habits or exercise habits, or because they might be on some medication (e.g. antidepressants) that have weakened their immune system.  That is, even assuming the causal link, you can't say that the causation would evaporate if the amputee declined the use of the prosthetic.
