Let's say there is a vaccine given to 2 year-olds that reduces the risk of contracting a disease by 50%.
The risk of an individual contracting the disease each year they are not vaccinated is 10%.
Thus the risk of the one vaccinated individual contracting the disease is 10% * 50% = 5% per year.
It takes about 29 years for an unvaccinated person to have a 95% chance of contracting the disease. (1-0.1)^29 < 0.05.
It takes 59 years for the vaccinated person to have a 95% chance of contracting the disease. (1-(0.5*0.1))^59 < 0.05.
Vaccines don't work like that, though.
That is the way circumcision as a STD preventive works, but vaccines are totally different.
Ok...there are two different flavors of a "50% effective" vaccine.
1) a vaccines which seroconverts 50% of the people who get it with one dose
2) a vaccine that reduced the symptoms/severity of the disease by 50%
Then there are mixtures of the two, but that's a lesser effect, so nevermind that for now.
With a vaccine that seroconverts (
effective seroconversion, to make it simple) 50% of people with one dose...let's assume it's an endemic childhood disease, and the average age of infection is age 5. Assuming vaccine uptake is low and the disease is still endemic, by age 10, your odds of having caught the disease (as in, got sick from the disease) are 50/50. Either way, you'll be immune (if you lived to tell the tale.) You either ran into the disease, the vaccine "worked" and you never got sick, and your immunity was "boosted" from the exposure...or the vaccine just failed, you caught the disease, and now you're immune from having had the disease. Either way, you'll probably be immune by age 10ish.
Make sense?
ETA:
And every time you run into the disease from there on out, you usually just become more and more immune (or if your immunity had started to wane, you're back to just "thoroughly immune"), without ever experiencing any symptoms and without passing the pathogen to someone else.
Your odds of eventually catching it don't go up as you get older unless mass vaccination has changed the average age of infection.