Question 3. [7 Marks] Pollution of the rivers in the United States has been a pr

Question

Question 3. [7 Marks] Pollution of the rivers in the United States has been a problem for many years. Consider the following probabilities: Stat 3502 A Assignment 1 Due In Tutorial a probability of 0.30 that a river is polluted a probability of 0.28 that a sample of water tested detects pollution, ·a probability of 0.22 that a river is polluted and a sample of water tested detects pollution, a probability of 0.03 that a river is polluted and fishing is permitted, a probability of 0.27 that a river is not polluted and fishing is permitted, a probability of 0.52 that a sample of water tested detects pollution or fishing is permitted and a probability of 0.02 that a river is polluted and fishing is permitted but a sample of water tested doesn't detect pollution. Let A, B and C be the events that a river is polluted, a sample of water tested detects pollution, and fishing is permitted, respectively. Find the probability that: a) [2 marks] fishing is permitted. b) 12 marks fishing is permitted but the sample tested does not detect pollution. c 3 marks] either a river is polluted, a sample of water tested detects pollution, or fishing is permitted

Explanation / Answer

here from above"

P(A) =0.30 ; P(B) =0.28 ; P(A n B)=0.22 ; P(A n C)=0.03 ; P(Ac n C)=0.27 ; P( B UC )=0.52

P(A n C n Bc)

a)

P(fishing is permitted)=P(C)=P(A n C)+ P(Ac n C) =0.03+0.27 =0.30

b)

here P(B n C)=P(B)+P(C)-P(B u C)=0.28+0.3-0.52=0.06

therefore P(fising is permitted but sample not detects pollution)

P(C n Bc) =P(C)-P(B n C)=0.3-0.06 =0.24

c)P(A n B n C) =P(A n C) -P(A n C n Bc) =0.03-0.02 =0.01

hence P(river polluted or detects pollution or fishing is permitted)

P(A u B u C)=P(A)+P(B)+P(C)-P(A n B)-P(A n C)-P(B n C)+P(A n B n C)

=0.3+0.28+0.3-0.22-0.03-0.06+0.01=0.58

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