Pollution of the rivers in the U.S. has been a problem for many years. Consider

Question

Pollution of the rivers in the U.S. has been a problem for many years. Consider the following events: A = {The river is polluted} B = {A sample of water detects the pollution} C = {Fishing permitted} Assume P(A) = 0.3, P(B|A) = 0.75, P(B|A') = 0.20, P(C|A Intersection B) = 02, P(C|A' Intersection B) = 0.15, P(C | A Intersection B') = 0.80, and P(C | A' Intersection B') = 0.90. a) Find P(A Intersection B Intersection c). b) Find P(B' Intersection C). c) Find P(C). d) Find the probability that the river is polluted given that the fishing is permitted and the sample tested did not detect pollution.

Explanation / Answer

P(A) = 0.3
P(A B) = P(A)P(B | A) = 0.3 × 0.75 = 0.225
P(A B C) = P(A B)P(C | A B) = 0.225 × 0.20 = 0.045

(b) Find P(BI C).
P(A I B) = P(A I)P(B | A I)

= (1 0.3) × 0.20

= 0.14
P(A I B C) = P(A I B)P(C | A I B)

= 0.14 × 0.15

= 0.021
P(A B I) = P(A) P(A B)

= 0.3 0.225

= 0.075
P(A B I C) = P(A B I)P(C | A B I)

= 0.075 × 0.80 = 0.06
P(B) = P(A B) + P(A I B)

= 0.225 + 0.14

= 0.365
P(A B) = P(A) + P(B) P(A B)

= 0.3 + 0.365 0.225

= 0.44
P(A I B IC) = P(A IB I)P(C | A IB I)

= (1P(AB))×P(C | A I BI)

=(1 0.44) × 0.90 = 0.504
P(B I C) = P(A B0 C) + P(A0 B0 C)

= 0.06 + 0.504 = 0.564

(c) Find P(C).
P(C) = P(B IC)+P(ABC)+P(A I BC)

= 0.564+0.045+0.021 = 0.630

(D) P(A | C B I) = (P (ABIC))/P (B IC)

=(0.06)/0.564 = 0.1064

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