Final answer:
The probabilities related to the presence of organisms in ballast water discharged from ships are determined using the Poisson distribution. Calculations address the likelihood of at least 8 organisms per cubic meter, exceeding the mean by more than one standard deviation in 1.5 m^3, and finding the discharge amount for a .999 probability of at least one organism.
Step-by-step explanation:
Probability of Organisms in Ballast Water
The probability that one cubic meter of ballast water discharged from a ship contains at least 8 organisms can be calculated using the Poisson distribution formula. The average concentration (mean, λ) is given as 10 organisms/m3. The probability P(X ≥ 8) is found by calculating 1 minus the sum of probabilities from 0 to 7 organisms. For part b, we assume that the number of organisms in 1.5 m3 is also Poisson distributed with a mean of 1.5 times that of 1 m3, which is 15. The standard deviation for a Poisson distribution is the square root of the mean, thus σ = √15. We find the probability that the number exceeds 15 + σ by computing 1 minus the probability of the number being less than or equal to this value. For part c, we solve for m in the equation P(X ≥ 1) = .999 with the mean λ = 10m, where m is the discharge amount in cubic meters.