Question

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(12 points) A genetic test is used to determine if people have a pred formation of a blood clot inside a blood vessel that obstructs the flow of system. It is believed tha person actualy has the predisposition actually has the predisposition iso 99. Thetest is 98%accurate if a person doesn hich is the predisposition for thrombosis, w of blood through the circulatory urate ta predisposition. The genetic test is 99%acc , meaning that the probability of a positive test resu not have the predisposition accurate if What is the probability that a randomly selected person who tests positive for the p actually has the predisposition? (A partially filled-in tree diagram is provided-you predisposition by the test want to fill in the restl) Predisposition for Thrombosis Test Results Positive Yes Negative No Positive Negative

Answer 1

Let D shows the evet that person has predisposition. So we have

$P(D)=0.02,P(D^{c})=1-P(D)=1-0.03=0.97$

Let P shows the event that test give positive result and N shows the event that test give negative result. So

$P(P|D)=0.99,P(N|D^{c})=0.98$

By the complement rule

$P(N|D)=1-P(P|D)=0.01,P(P|D^{c})=1-P(N|D^{c})=0.02$

By the law of total probability, the probability that test gives positive results is

$P(P)=P(P|D)P(D)+P(P|D^{c})P(D^{c})=0.99\cdot 0.03+0.02\cdot 0.97=0.0491$

By the law of probability, the probability that person actually have predisposition given that test results positive is,

$P(D|P)=\frac{P(P|D)P(D)}{P(D)}=\frac{0.99\cdot 0.03}{0.0491}=0.6049$