Quote:
Originally Posted by erimir
Whether that's really bad depends on the vaccination rate.

Quote:
Originally Posted by But
Quote:
Originally Posted by godfry n. glad
So in any random exposed population, how many vaccinated individuals would you expect to develop the infection?

Obviously, that depends on the percentage of vaccinated people in that population. Let's say 91% are vaccinated, the vaccine has a failure rate of 5% and 90% of unvaccinated people get the disease.
Then
P(sick  vaccinated) = 5%.
P(vaccinated  sick) = P(vaccinated and sick) / P(sick) = 0.91 * 0.05 / (0.91 * 0.05 + 0.09 * 0.9) = (approx.) 0.36 = 36%
If you assume that everyone is vaccinated and the vaccine has a nonzero failure rate, you would get that 100% of those infected were vaccinated.

I didn't feel like doing the math or really explaining what I meant, but But apparently had already explained it.
This is similar to the difference between sensitivity and specificity and similar measures in terms of medical tests.
Given a person who has the condition, the likelihood the test will return a positive (sensitivity)
Given a person who does not have the condition, the likelihood the test will return a negative (specificity)
Given a person who has tested positive, the likelihood the person indeed has the condition (positive predictive value)
Given a person who has tested negative, the likelihood the person indeed does not have the condition (negative predictive value)
These are very different numbers. Sensitivity and specificity are not affected by the condition's prevalence in the population, while predictive values are. If you test a random person in the US for HIV, and you get a positive, the person still probably does not have HIV. If you test a random person in Botswana, that would not be the case, because the infection rate is much higher there.
Vaccines obviously are not the same as diagnostic tests, but the mathematical relationships are similar. The failure rate is a measure solely related to the vaccine, but the percentage of infected who are vaccinated is strongly affected by the vaccination rate in the population.
To give the sort of opposite of slimshady's point  suppose that only a hundred people in France were vaccinated, and
all of them caught measles, while a million people overall were infected. That would mean 99.99% of the infected were unvaccinated and 0.01% were vaccinated. Would you conclude from that that the vaccine is 99.99% effective? That would clearly be ludicrous.