Question

Two important indicators of stream pollution are high biological oxygen demand (BOD) and low pH. Of the more than 250 streams draining into a large lake, 30% have high BOD and 20% have low pH levels, with 10% having both characteristics.

Answer 1

Answer:

a) The Venn diagram is presented in the attached image to this answer.

b) Check Explanation.

c) 0.3333

d) 0.1429

e) 0.6

Explanation:

Let the probability of a lake having high BOD be P(B) = 30% = 0.3

Probability of a lake having low pH = P(P) = 20% = 0.2

Probability that a lake has high BOD and low pH = P(B n P) = 10% = 0.1

Then, probability that a lake has normal BOD = P(B') = 1 - P(B) = 1 - 0.3 = 0.7

Probability that a lake has normal pH = P(P') = 1 - P(P) = 1 - 0.2 = 0.8

Total probability = P(U) = 100% = 1

a) The Venn diagram is presented in the attached image to this answer.

b) Two events are independent if and only if, P(A|B) = P(A) or P(B|A) = P(B).

For this question,

P(B|P) = P(B n P)/P(P) = 0.1/0.2 = 0.5 ≠ P(B) (which is 0.3)

And P(P|B) = P(B n P)/P(B) = 0.1/0.3 = 0.333 ≠ P(P) (which is 0.2).

It is evident that the two events aren't independent of each other.

c) If a stream has high BOD, what is the probability it will also have low pH?

This probability is given as P(P|B) meaning that, the probability of a lake having low pH given that it has high BOD.

Mathematically, this conditional probability is given by

P(P|B) = P(B n P)/P(B) = 0.1/0.3 = (1/3) = 0.3333

d) If a stream has normal levels of BOD, what is the probability it will also have low pH.

This probability is given as P(P|B'); that is, the probability of a lake having low pH given that it has normal BOD.

Mathematically,

P(P|B') = P(B' n P)/P(B')

P(B') = 0.7 (already found above)

But P(B' n P) = ?

Mathematically,

P(B' n P) = P(P) - P(B n P) = 0.2 - 0.1 = 0.1

P(P|B') = 0.1/0.7 = 0.1429

e) What is the probability that a stream will not exhibit either pollution indicator, i.e., will have normal BOD and pH levels?

This is given as P(B' n P')

Mathematically, this represents the region in the Venn diagram outside of the circles representing P(B) and P(P) and it's given mathematically as,

P(B' n P') = P(U) - [P(B n P') + P(B' n P) + P(B n P)] = 1 - (0.2 + 0.1 + 0.1) = 1 - 0.4 = 0.6 or 60%