Problem #3: (15pts) Consider two events X and Y with probabilities, PO) -7/1s, P

Question

Problem #3: (15pts) Consider two events X and Y with probabilities, PO) -7/1s, Px-1/3, and PX1D) 2/3. Calculate P(), Pr/X), and P(Y/X). State with reasons whether the events X and Y are dependent or mutually exclusive. Problem 2.93 from textbook (6th edition): (10pts) The following table summarizes the number of deceased beetles under autolysi after its death by the action of i especially protein, by microorganisms, resulting in production of foul-smelling matter): s (the destruction of a cell ts own enzymes) and putrefaction (decomposition of organic matter Autolysis Low 47 14 High Low Putrefaction (a) If the autolysis of a sample is high, what is the probability that the putrefaction is low? (b) If the putrefaction of a sample is high, what is the probability that the autolysis is high? (c) If the putrefaction of a sample is low, what is the probability that the autolysis is low?

Explanation / Answer

1) P(Y) = P(X Y) / P(X | Y) = (1/3) / (2/3) = 1/2 (ans)

P(Y | X) = P(X Y) / P(X) = (1/3) / (7/15) = 5/7  (ans)

P(X' Y) = P(Y) - P(X Y) = 1/2 - 1/3 = 1/6

P(Y | X') = P(X' Y) / P(X') = (1/6) / (8/15) = 5/16  (ans)

2) Total = 14 + 30 + 47 + 9 = 100

a) P(putrefaction is low | autolysis is high) = P(putrefaction is low and autolysis is high) / P(autolysis is high)

= (30/100) / ((14 + 30) / 100)

= 30 / 44

= 0.6818

b) P(autolysis is high | putrefaction is high) = P(autolysis is high and putrefaction is high) / P(putrefaction is high)

= (14/100) / ((14+47)/100)

= 14/61

= 0.2295

c) P(autolysis is low | putrefaction is low ) = P(autolysis is low and putrefaction is low) / P(putrefaction is low)

= (9/100) / ((30+9)/100)

= 9/39

= 0.2308

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