Question

the number of fatalities due to shark attack during a year is modeled using a poisson distribution. the international shark attack file (isaf) investigates shark-human interactions worldwide. inter- nationally, an average of 4.4 fatalities per year occurred during a 5-year period. assuming that this mean remains constant for the next 5 years, find the probabilities of the following events. a no shark fatalities will be recorded in a given year. b sharks will cause at least six human deaths in a given year. c no shark fatalities will occur during the 5-year period. d at most 12 shark fatalities will occur during the 5-year perio

Answer 1

To find the probabilities in this problem, we use the Poisson distribution formula. The probabilities are as follows: a) approximately 0.012, b) approximately 0.505, c) approximately 0.003, d) approximately 0.005.

Step-by-step explanation:

To calculate the probabilities in this problem, we will use the Poisson distribution formula. Let's define the following variables:

x = number of fatalities

a) To find the probability of no shark fatalities in a given year, we plug x = 0 into the Poisson distribution formula. P(X = 0) = (e^(-λ) * λ^0) / 0! = (e^(-4.4) * 4.4^0) / 0! ≈ 0.012.

b) To find the probability of at least six human deaths in a given year, we calculate the cumulative probability from x = 6 to infinity. P(X >= 6) = 1 - P(X <= 5). We can calculate P(X <= 5) using the Poisson distribution formula with x = 5. P(X <= 5) = ∑[k=0 to 5] (e^(-λ) * λ^k) / k! ≈ 0.495. Therefore, P(X >= 6) = 1 - 0.495 ≈ 0.505.

c) To find the probability of no shark fatalities during the 5-year period, we again use the Poisson distribution formula with the average number of fatalities for the 5-year period, which is λ * 5. P(X = 0) = (e^(-λ*5) * (λ*5)^0) / 0! ≈ 0.003.

d) To find the probability of at most 12 shark fatalities during the 5-year period, we calculate the cumulative probability from x = 0 to 12. P(X <= 12) = ∑[k=0 to 12] (e^(-λ*5) * (λ*5)^k) / k!. This calculation would involve adding up 13 terms, which can be time-consuming.

Alternatively, we can use the complement rule: P(X <= 12) = 1 - P(X > 12). We can calculate P(X > 12) using the cumulative probability from x = 0 to 11. P(X <= 11) = 0.995 (approximately). P(X > 12) = 1 - 0.995 = 0.005 (approximately).