3. Continue on the medical testing example in Q2. We know that 1 in 2000 people

Question

3. Continue on the medical testing example in Q2. We know that 1 in 2000 people in the population is a carrier of the disease, i.e. P(A)=0.0005. The probability that a carrier tests negative is 0.01, i.e. P(B/A^c)=0.01. The probability that a non-carrier tests positive is . 02, i.e.P(B/A^c) =0.02. (a) Calculate the probability that the patient is actually a carrier given the patient is tested positive. (b) Calculate the probability that the patient is actually NOT a carrier given the patient is tested negative. (c) Suppose there are grave consequences if the carrier misses the treatment, while the treatment on non-carrier has very small negative effects. If you are the doctor, what would you do to improve the confidence of the testing procedure, i.e. would you reduce the likelihood of false positive or the likelihood of false negative? Use Bayes Theorem to justify your reason

Explanation / Answer

P(A) = 0.0005

P(Bc | A) = 0.01 =>  P(B | A) = 1 - P(Bc | A) = 1- 0.01 = 0.99

P(B | Ac) = 0.02 => P(Bc | Ac) = 1 - 0.02 = 0.98

(a)

P(A | B)

= P(A) P(B |A) / [ P(A) P(B |A) + P(Ac) P(B |Ac) ]

= 0.0005 * 0.99/ (0.0005 * 0.99 + (1-0.0005) * 0.02)

=0.024

(b)

P(Ac | Bc)

= P(Ac) P(Bc |Ac) / [ P(A) P(Bc |A) + P(Ac) P(Bc |Ac) ]  

= (1- 0.005) *0.98 / ((1- 0.005) *0.98+ 0.005*0.01)

= 0.9999

(c)

P(B | AC) = 0.02 ( false positive , is not that serious)

P( Bc | A ) = 0.01 (false negative is serious )

Hence we must reduce the likelihood of false negative.

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