That's a great argument for persuading someone in a population where measles is rampant that they need to get vaccinated (or get their kids vaccinated). The US in the 1950s, for example, was such a population. 95% of the population got measles by age 15 [1].

Given the numbers you give [2], as long as the measles rate is above 2500 per 100k in your population the probability of going deaf from measles in a non-vaccinated person is higher than the probability of something really bad happening to that person from vaccination.

For persuading someone in a population where measles is rare due to past eradication efforts and herd immunity it's not so great an argument. In the US, we are way below 2500 per 100k.

To make a persuasive case using that approach you are going to need to pull in all the possible bad outcomes of measles. Deafness, death, and whatever else it can do. That might do it.

The ideal way to deal with a diseases is universal vaccination (except for people who for medical reasons cannot be safely vaccinated) until the disease is completely eradicated, then stop vaccination people.

Near the end of that, though, you reach a stage where the disease has not been completely eradicated but is rare enough that for any given individual skipping vaccination may actually be lower risk for them personally provided that enough other people do get vaccinated.

At that point, you have to switch to arguments based on the good of the group as a whole instead of personal risk to individuals.

[1] https://www.ncbi.nlm.nih.gov/pubmed/6878996

[2] I'm assuming 1:N means odds are 1 to N, meaning a probability of 1/(N+1).

To be very clear, I fully support vaccination, my kids are vaccinated, as am I.

Having said that, when people hear the risk of a bad reaction is 1:40000, I believe that what they think is "my kid's chances of getting this disease are lower than that, so I am making the smart choice by picking the lower risk and avoiding the vaccine risk".

And in some cases that's probably the rational position; even then it's still rational (and right) for the medical community to advise vaccination as they "don't care" about that one person, they care about the billions.

This dichotomy lies at the heart of this issue IMO.

By those numbers, if she thinks the chance of getting measles is less than 1/40, then if acting purely selfishly she'd prefer not to get the vaccine?

No. If you hadn't added "she thinks" to your sentence you would be correct - but the if condition is false (the risk is worse than 1/40) and so she should get the vaccine.

Where are you getting the measles risk from?

All the numbers I can find put the risk of measles in the US at a couple of orders of magnitude lower than 1/40. About 1000 cases a year in a bad year, 320 million population, 80 year lifespan give about a 1/4000 chance of getting measles sometime over your lifespan.

This is complex. Measles risk depends on how many other people are vaccinated. Right now the US is above herd immunity rates meaning that for everybody the risk is insignificant. However we are close to a tipping point, if vaccination rates go much lower the risk for everybody increases dramatically (but how much I don't know). Note that risk for everybody is just that: for some vaccinated people the vaccination doesn't work (As I recall it is about 95% works) - the risk is drastically higher for the non-vaccinated of course.

If it is only one person in isolation asking this question the risk of vaccines is worse than the risk of the disease - but if more than one person asks this questions the risk of measles for everybody is higher.

There is also the risk that some mutation of the virus will evade the current vaccine. I don't know how big that risk is (the flu mutates such that a new vaccine is needed yearly), at which point everybody is vulnerable. If enough people are vaccinated the disease has many less opportunities for this and will die.