Question 2 (Independence and Conditional Probabilities)
A river downstream from farmland is monitored daily for both nitrogen and phosphorus concentrations. Higher concentrations of nitrogen and phosphorus indicate greater pollution of the water. Specifically, the days when the nitrogen concentration N in the water exceed a critical nitrogen threshold concentration [N_(critical )] are recorded, and the days when the phosphorus concentration P in the water exceed a critical threshold phosphorus concentration [P_(critical )] are recorded. On any given day, the probability that [N]>[N_(critical )] is 20%, and the probability that [P]>[P_(critical )] is 15%. The probability that both nitrogen and phosphorus concentrations exceed their critical thresholds, Pr(|N|)>|N_(critical )|\cap |P|>(|P_(critical )|), is 5%.
a) On any given day, calculate the probability that both the nitrogen and phosphorus concentrations are below their critical threshold concentrations.
b) If it is known that on a given day the phosphorus concentration exceeds its critical threshold, calculate the probability that the nitrogen concentration will also exceed its critical threshold.
c) Are the events |N|>|N_(critical )| and |P|>|P_(concal )| independent, mutually exclusive, or neither? Explain your auswer.
