Question

2. a disease called mathitosis is sweeping through the college campus's 10,000 students . 15% of campus is infected. a test has been developed, that is 90% accurate at detecting the disease, but has 17% false positive result .

2a) zuzana tested postive , what is the probability she has the disease?

2b) rafael tested negative , whats the probability he actually has the disease?

Answer 1

Consider this tree diagram,

Let X denote whether the student is diseased or not diseased.

let Y denote whether the student tests positive or negative.

A test has been developed, that is 90% accurate at detecting the disease, but has 17% false positive result. Then obviously it has 83% true positive result. Hence, the tree diagram.Y=P

P[X=D]= 0.9, P[X=ND]=0.1

P[Y=P|X=D]= 0.83, P[Y=P|X=ND]= 0.17

P[Y=N|X=D]=0.17, P[Y=N|X=ND]= 0.83

a) zuzana tested postive , what is the probability she has the disease?

we need to find P[X=D|Y=P]

P[Y=P]= P[Y=P|X=D] + P[Y=P|X=ND] = 1

P[X=D, Y=P] = 0.9*0.83 = 0.747

So, P[X=D|Y=P]= 0.747

b) Rafael tested negative , whats the probability he actually has the disease?

we need to find P[X=D|Y=N]

P[Y=N]= P[Y=N|X=D] + P[Y=N|X=ND]= 0.17+ 0.83= 1

P[X=D, Y=N]= 0.9*0.17= 0.153

So, P[X=D|Y=N]= 0.153