# Is it reasonable to expect any process to have a zero error rate ?

I am interested in the philosophical answer to the question: can any complex process with a measurable outcome of success or failure truly have a zero error rate (ie 100% success and 0% failure). I accept that that one can measure outcomes of processes and usually observe a non zero error rate, though if you do observe a non zero error rate can you say with certainty that this is a sampling error.

As an example:

A health insurance company in Australia is proposing that it will not pay hospitals when certain adverse events happen (post operative infection or blood clots and about 100 other events) on the basis that these are considered 'highly preventable' and the observance of such events should be considered a 'mistake'.

It is true that applying certain preventative measures (prophylactic antibiotics or blood thinning agents) will reduce the risk of these relatively uncommon events though the relative risk reduction is moderate. Each of the listed events have an observed non zero rate of occurrence when all known prophylactic treatments are applied. The health insurers proposal is not to withhold payment if prophylactic measures are withheld but to withhold payment if the adverse event occurs - regardless of anything else. I understand that it should be sufficient to argue that non zero error rates are observed in optimal circumstances in all of the situations proposed, though I want to know if it can be argued in general that insistence on a a non zero error rate is asking for the impossible.

• You might want to consider what light a simple Bayesian calculation might shed on this circumstance, because the rates to focus on are not the raw rates of observing these events, nor the rates at which such events result from mistakes, but rather the chance that such an event, if observed, actually resulted from a mistake. Bayes' Theorem tells us this chance depends not only on the rates at which mistakes are made but also on the rates at which such events are produced by non-mistakes. – whuber Aug 14 '15 at 22:32
• This is relevant: stats.stackexchange.com/questions/82720/… An confidence interval for $p$ with zero successes. – kjetil b halvorsen Aug 17 '15 at 9:12

In this world nothing can be said to be certain, except death and taxes."

                                 --Ben Franklin


If an event has any non-zero probability of occurring, then it will almost surely happen if you keep doing it. As @mandata pointed out above, you can use the geometric distribution to calculate the probability of seeing one "success" (here, a medical failure) in $k$ trials. Alternately, it's easy to show that the expected number of trials needed for one success is $1/p$.

The Center for Disease Control estimates that the surgical site infection rate was around 2%. I have no idea how effective antibiotic prophylaxis is, but I am certain that post-op antibiotics cannot drive the infection rate to zero. Even if we assume that they're highly effective and reduce the infection rate by 100-fold, we'd expect an infection after 5,000 surgical sites, which doesn't seem like very many for a big hospital.

However, the insurer isn't exactly demanding a zero error rate. It might be fairer to think of them as 'fining' the hospital the cost a procedure for each adverse event to encourage them to take steps to prevent post-operative infections, which are a pretty substantial cause of death. More cynically, you could also accuse them of cutting their reimbursement rates by $p \cdot \textrm{price}$ dollars.

Even services that are largely automated and easier to test, backup, etc. don't have 0% error rates, though they're close. Amazon's cloud computing platform was up for 99.9974% of 2014.

Given that the probability of an event is constant and/or is greater than a fixed lower-limit, with a sufficient number of trials, the event will occur with probability 1. This can be seen with something as simple as the CDF of the geometric distribution, where $lim_{k\rightarrow \inf} 1-(1-p)^k \rightarrow 1$.

• This doesnt answer the question. – kjetil b halvorsen Aug 17 '15 at 9:06
• The questioner states, "Each of the listed events have an observed non zero rate of occurrence when all known prophylactic treatments are applied." For an event to have a zero error rate, the probability has to go to zero. That is not the case here. By zero-one law the event will occur with certainty. Are "they asking for the impossible?" Yes. – mandata Aug 17 '15 at 13:38
• @kjetilbhalvorsen, I think it does a pretty decent job though it may be a little technical for the OP. – Matt Krause Aug 19 '15 at 16:22