Question 1087127: A certain virus and fix one in every 200 people. A test used to detect a virus in a person is positive 70% of the time with the person has the virus and 5% of the time with the person does not have the virus. This 5% is called a false positive. Let a be the event the person is infected it be be there event the person test positive.
A) Using Bayes Theorem, when a person tests positive, determine the probability that the person is infected.
B) Using Bayes Theorem, when a person tests negative, determine the probability that the person is not infected.
Answer by mathmate(429)   (Show Source):
You can put this solution on YOUR website! Question:
A certain virus and fix one in every 200 people. A test used to detect a virus in a person is positive 70% of the time with the person has the virus and 5% of the time with the person does not have the virus. This 5% is called a false positive. Let a be the event the person is infected it be be there event the person test positive.
A) Using Bayes Theorem, when a person tests positive, determine the probability that the person is infected.
B) Using Bayes Theorem, when a person tests negative, determine the probability that the person is not infected.
Solution:
Events:
V=infected with Virus
~=not infected
P=tests Positive
~P=tests negative
Given:
P(V)=0.005
P(P|V)=0.7
P(P|~V)=0.05
Calculations:
using total probability
P(P)=percentage testing positive
=P(P|V)*P(V)+P(P|~V)*P(~V)
=0.7*0.005+0.05*0.995
=0.05325
(a) when a person tests positive, determine the probability that the person is infected.
P(V|P)=P(V∩P)/P(P)
=P(P∩V)/P(P)
=P(P|V)*P(V)/P(P)
=0.7*0.005/0.05325
=0.06573
(b) when a person tests negative, determine the probability that the person is not infected.
P(~V|P)=P(~V∩P)/P(P)
=P(P∩~V)/P(P)
=P(P|~V)*P(~V)/P(P)
=0.05*0.995/0.05325
=0.93427
Check: 0.06573+0.93427=1.00000 ok
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