A patient comes into a doctor office exhibiting two symptoms: s1 and s2. The doctor has two possible diagnoses: disease d1 or disease d2. Assume that, given the symptoms, the patient must have either d1 or d2, but cannot have both. The following probabilities are given:
P(s1|d1) = 0.8
P(s1|d2) = 0.4
P(s2|d1) = 0.2
P(s2|d2) = 0.6
P(d1) = 0.003
P(d2) = 0.007
Assume that s1 and s2 are conditionally independent, given the disease.
1) What are P(d1|s1,s2) and P(d2|s1,s2)?
Answer:Code:
P(d1|s1,s2) = P(s1|d1)P(s2|d1)P(d1)/P(s1,s2)
= P(s1|d1)P(s2|d1)P(d1)/(P(s1|d1)P(s2|d1)P(d1) + P(s1|d2)P(s2|d2)P(d2))
= (0.8)(0.2)(0.003)/((0.8)(0.2)(0.003)+(0.4)(0.6)(0.007))
= 0.22
P(d2|s2,s2) = P(s1|d2)P(s2|d2)P(d2)/P(s1,s2)
= P(s1|d2)P(s2|d2)P(d2)/(P(s1|d1)P(s2|d1)P(d1) + P(s1|d2)P(s2|d2)P(d2))
= (0.4)(0.6)(0.007)/((0.8)(0.2)(0.003)+(0.4)(0.2)(0.007))
= 0.78
= 1 - P(d1|s1,s2)
2) The doctor has the choice of two treatments t1 and t2. (It is not an option to do both.) Let c be the event that the patient is cured. The following probabilties are given:
P(c|d1,t1) = 0.8
P(c|d2,t1) = 0.1
P(c|d1,t2) = 0.3
P(c|d2,t2) = 0.6
Assume that event c is conditionally independent of the symptoms, given the disease and the treatment.
What is the P(c|t1,s1,s2)?I have been working on this problem for a week and I still couldn't solve question 2? Any tips would be much apprecipated..
Login
FAQ
Search
