# How to use the word "Knowing" in statistics

A vaccine has a 90% probability of preventing a certain disease. The probability of getting this disease if a person is not vaccinated is 50%. In a certain region, 25% of people get vaccinated against this disease. Find the probability that a randomly selected person from this region

A. has the disease and was not vaccinated.
B. doesn't have the disease and was vaccinated.
C. has the disease.
D. wasn't vaccinated knowing that he/she has the disease

My attempt:
using a probability tree i made;
where V=Vaccinated (V^c = not vaccinated), D=has disease (D^c= doesn't have disease)

a) p(V^c)P(D|V^c)
=.75*.50
=0.375

b)P(V)P(D^c|V)
=0.25*0.90
=0.225

c) P(V)P(D|V)+P(V^c)P(D|V^c)
=(0.25*0.10)+(0.75*0.50)
=0.4

d) this is where i have the issue, what does the word Knowing mean? i assumed that i would just go ahead and do
P(V^c)P(D|V^c)
=0.75*0.50
=0.375
but since it says knowing, im not sure if there is something else i have to do because they know they have the disease, maybe instead it would be 0 since they wont get vaccinated now that they have the disease?

[Moderator's note: Moved from a technical forum and thus no template.]

## Answers and Replies

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Mark44
Mentor
D. wasn't vaccinated knowing that he/she has the disease
I interpret this to mean that the subject didn't get vaccinated because he/she already has the disease. A vaccination is supposed to prevent you getting its associated disease. If you know you have the disease, what's the point in getting a vaccination for it?

Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
I interpret this to mean that the subject didn't get vaccinated because he/she already has the disease. A vaccination is supposed to prevent you getting its associated disease. If you know you have the disease, what's the point in getting a vaccination for it?
This is not my interpretation. The question is more likely intending to ask for the probability that a given person who has fallen ill was vaccinated. In other words: ”what is the probability that a person is vaccinated conditioned on that person having fallen ill”.

PS. This is a stupid question as it fails to account for population effects such as herd immunity.

Stephen Tashi
Science Advisor
D. wasn't vaccinated knowing that he/she has the disease
d) this is where i have the issue, what does the word Knowing mean? i assumed that i would just go ahead and do
P(V^c)P(D|V^c)
A more common way to phrase the question in English is: What is the probability the selected person was not vaccinated given that he/she has the disease? This would make it clear that the student working the problem knows the person has the disease - instead of suggesting that the person selected knows that fact.

Your answer computed $P(V^C \cap D)$ , however question d) asks you to find $P(V^C| D)$

On the site you mentioned in another thread: https://www.zweigmedia.com/RealWorld/tutorialsf15e/frames7_5.html

You can find that $P(E|F) = \frac{ P(E \cap F)}{P(F)} = \frac{ P(F | E) P(E)}{P(F)}$

You can apply that rule to find $P(V^C | D)$

A more common way to phrase the question in English is: What is the probability the selected person was not vaccinated given that he/she has the disease? This would make it clear that the student working the problem knows the person has the disease - instead of suggesting that the person selected knows that fact.

Your answer computed $P(V^C \cap D)$ , however question d) asks you to find $P(V^C| D)$

On the site you mentioned in another thread: https://www.zweigmedia.com/RealWorld/tutorialsf15e/frames7_5.html

You can find that $P(E|F) = \frac{ P(E \cap F)}{P(F)} = \frac{ P(F | E) P(E)}{P(F)}$

You can apply that rule to find $P(V^C | D)$
oh! alright so the answer should be

P(V^c ∩ D) / (P(D))
=0.375/0.4
=0.9375
Is this correct?

Stephen Tashi
Science Advisor
=0.9375
Is this correct?
Yes

Mark44
Mentor
wasn't vaccinated knowing that he/she has the disease
PS. This is a stupid question as it fails to account for population effects such as herd immunity.
It's also a poorly formed question in that it is not clear who the subject is in the clause "knowing that he/she has the disease".

HallsofIvy
Science Advisor
Homework Helper
I am going to step in with my standard method of solving problems like this, not using formulas.

Imagine a population of 1000 people. 25% of them, 250, get vaccinated. 90% of those, 225 do not get the disease, the other 10%, 25 people, do get the disease. Of the 750 people who do not get vaccinated, 50%, 375 get the disease, the other 375 do not.

Find the probability that a randomly selected person from this region
A. has the disease and was not vaccinated.
That is 375 people out of the original 1000 so is 0.375 as you say.

B. doesn't have the disease and was vaccinated.
That is 225 out of the original 1000 so is 0.225 as you say.

C. has the disease.
A total of 25+ 375= 400 out of the original 1000 so 0.4 as you say.

D. wasn't vaccinated knowing that he/she has the disease
A total 25+ 375= 400 people got the disease and 375 of those were not vaccinated so 375/400= 0.9375, again, exactly what you have. Well done!