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ECE109 Disc3.pdf

ECE109 Disc3.pdf - UC San Diego J Connelly ECE 109...

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UC San Diego J. Connelly ECE 109 Discussion 3 Notes Problem 3.1 Suppose you are a farmer, and 25% of your plants have a disease. The best available test detects the disease 90% of the time when it is present. However, the test yields false positives 10% of the time. (a) If a plant tests positive, what is the probability is has the disease? (b) You discover the disease is 50% fatal when it is present, the treatment completely eradicates the diseases, but the treatment has a 5% fatality rate. What percentage of your plants should you expect to survive when: (i) you do not treat any of your plants? (ii) you treat all of your plants? (iii) you treat the plants that test positive? Solutions (a) Let A be the event of a plant tests positive, and let X be the event the disease is present. We have P ( X )= 1 4 , P ( A | X )= 9 10 , P ( A | X c )= 1 10 P ( X | A )= P ( AX ) P ( A ) = P ( A | X ) P ( X ) P ( A ) = P ( A | X ) P ( X ) P ( AX )+ P ( AX c ) = P ( A | X ) P ( X ) P ( A | X ) P ( X )+ P ( A | X c ) P ( X c ) = 9 10 1 4 9 10 1 4 + 1 10 3 4 = 3 4 =75% (b) Let L the event a plant lives. (i) When no treatment is given, we have P ( L | X )= 1 2 and P ( L | X c )=1 , so P ( L )= P ( LX )+ P ( LX c )= P ( L | X ) P ( X )+ P ( L | X c ) P ( X c ) = 1 2 1 4 +1 3 4 = 7 8 =87 . 5% (ii) When treatment is given to every plant, we have P ( L )= 19 20 =95% (iii) When treatment is given to the plants that test positive P ( L | AX )= P ( L | AX c )= 19 20 , P ( L | A c X c )=1 , and P ( L | A c X )= 1 2 so P ( L )= P ( LAX )+ P ( LAX c )+ P ( LA c X )+ P ( LA c X c ) = P ( L | AX ) P ( A | X ) P ( X )+ P ( L | AX c ) P ( A | X c ) P ( X c ) + P ( L | A c X ) P ( A c | X ) P ( X )+ P ( L | A c X c ) P ( A c | X c ) P ( X c ) = 19 20 9 10 1 4 + 19 20 1 10 3 4 + 1 2 1 10 1 4 +1 9 10 3 4 = 389 400 =97 . 25% Please report any typos/errors to [email protected]
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