Statistics Probability question

#1
At an allergist’s office, patients are screened for having a peanut allergy. The probability of having a peanut allergy is 0.023. The probability that the test comes up positive for those with a peanut allergy is 0.80. The probability that the test provides a negative result for those who do not have the allergy is 0.90.

a. What’s the probability of having a peanut allergy?
P (peanut allergy) = 0.023

b. Find the probability that the test results in a positive outcome
P(+ result): P(+ result/allergy) + P(+ result/no allergy) = (0.023x0.80) + (0.977x0.10) = .1161

c. Given that the test results in a positive outcome, what is the probability of having a peanut allergy?
P(+ result/allergy) = P(+result / allergy) / P(+result) = .0184/.1161 = .1585

This is what I have so far, but the last time I did a similar problem, my teacher said that part c. is not conditional, and that you have to multiply across the branches. I'm not sure how to do that part. What is the correct way to do part c.? Thanks!
 

Dason

Ambassador to the humans
#2
You're really interested in P(allergy | +result) - not P(+result | allergy). Have you talked about Bayes' theorem?
 
#5
It's a theorem describing how the conditional probability of a set of possible causes for a given observed event can be computed from knowledge of the probability of each cause and the conditional probability of the outcome of each cause but I still don't understood how to use that in this problem. Would you use 0.023, which is the probability of having a peanut allergy, in part c?
 

Dason

Ambassador to the humans
#6
I suggest that you actually write out what Bayes' theorem says symbolically. Then write out all of the information that you do know. Then write out what you want to know. See if you can use Baye's theorem to rewrite what you want to know in terms of things that you do know.

Keep in mind that the thing you want to know is P(allergy | +result).
 
#7
I suggest that you actually write out what Bayes' theorem says symbolically. Then write out all of the information that you do know. Then write out what you want to know. See if you can use Baye's theorem to rewrite what you want to know in terms of things that you do know.

Keep in mind that the thing you want to know is P(allergy | +result).

I'm not sure if its right but this is what I have so far: P(allergy | +result) = P(0.023) x P(.0184) / P(0.023) x P(0.0184) + P(0.023)..
I know you have to square it to something but I don't know what..