A recent study to determine the efficacy of a diagnostic tool consisted of a sam

Question

A recent study to determine the efficacy of a diagnostic tool consisted of a sample of 400 patients. Three hundred were known to be positive. A total of 280 tested positive. Of the patients that tested positive, 250 were known positive.

Let T be the event that a patient tests positive

Let K be the event that a patient is known to be positive

a) Show the joint probability table

b) Are T and K independent events? Explain.

c) What is the probability that a patient that tests negative is actually negative? What is the probability that a patient tests negative, when we know she/he is actually negative? Explain the difference.

Explanation / Answer

Ans:

a)

b)For T and K to be independent,

P(T and K)=P(T)*P(K)

Check:

P(T and K)=0.625

P(T)=0.75*0.7=0.525

Both are not equal,so T and K are not independent.

c)P(K'/T')=P(T' and K')/P(T')=0.175/0.3=0.583

P(T'/K')=P(T' and K')/P(K')=0.175/0.25=0.7

T T' Total K 250 50 300 K' 30 70 100 Total 280 120 400 Joint probability table T T' Total K 0.625 0.125 0.75 K' 0.075 0.175 0.25 Total 0.7 0.3 1

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