Thread: Probability / Flu Vaccinations

I'm given that 70% of residents in a nursing home are given the flu vaccination, and a doctor observes that 30% of those who have the flu have already been vaccinated against it. I am trying to find the probability that a resident gets the flu, given that they have been vaccinated against it, and then also the chance they get the flu without being vaccinated.
What I set up:
P(V) = .7
P(Vc) = .3
P(V|F) = .3
and I am trying to find:
P(F|V)
P(F|Vc)

I tried using the definition of conditional probability, so that P(F|V) = P(F intersect V) / P(V), but I can't figure out what P(F intersect V) is because I only know P(V). Can anyone point me in the right direction? Thanks!

Originally Posted by mistykz

I'm given that 70% of residents in a nursing home are given the flu vaccination, and a doctor observes that 30% of those who have the flu have already been vaccinated against it. I am trying to find the probability that a resident gets the flu, given that they have been vaccinated against it, and then also the chance they get the flu without being vaccinated.
What I set up:
P(V) = .7
P(Vc) = .3
P(V|F) = .3
and I am trying to find:
P(F|V)
P(F|Vc)

I tried using the definition of conditional probability, so that P(F|V) = P(F intersect V) / P(V), but I can't figure out what P(F intersect V) is because I only know P(V). Can anyone point me in the right direction? Thanks!

Can't be done. This is easily illurstrated by supposing 100 residents, 70 of which are vacinated.

Suppose there are 10 flu cases 3 of which have been vacinated. This is consistent with all the given data.

Now suppose there are 20 flu cases 6 of which have been vacinated. This is also consistent with all the given data.

The conditional probability of getting flue given vacination in the first case is 3/70, and in the second 6/70.

CB

What I think you have is...

P(V)=.7 and P(V|F)=.3
where V= vaccinated and F=Flu

You need P(F) to finish this

$\displaystyle P(F|V)={P(FV)\over P(V)}={P(V|F)P(F)\over P(V)}$

and

$\displaystyle P(F|V')={P(FV')\over P(V')}={P(F)-P(FV)\over 1-P(V)}$

so this can't be finished at all?

Originally Posted by mistykz

so this can't be finished at all?

Not without more data.

CB