91Ó°ÊÓ

The Poisson Probability Formula is given by: \(P(k; λ) = \frac{e^{−λ} \cdot λ^{k}}{k!}\) Where: - P(k; λ) is the probability of observing k events in a given period, - λ is the average number of events in the period, and - k is the number of events we want the probability for.

(a) More than 2 accidents in the next month: To find the probability of more than 2 accidents, we need to sum the probabilities of 3 accidents, 4 accidents, and so on. However, this can be quite difficult since it can involve an infinite number of terms. Instead, we can use the complement rule: \(P(A') = 1 - P(A)\), where \(P(A')\) is the probability of more than 2 accidents and \(P(A)\) is the probability of 2 or fewer accidents. We can compute \(P(A)\) as follows: \(P(A) = P(0; 2.2) + P(1; 2.2) + P(2; 2.2)\) We will calculate each of these probabilities using the Poisson Probability Formula: \(P(0; 2.2) = \frac{e^{−2.2} \cdot 2.2^{0}}{0!} \approx 0.1108\) \(P(1; 2.2) = \frac{e^{−2.2} \cdot 2.2^{1}}{1!} \approx 0.2438\) \(P(2; 2.2) = \frac{e^{−2.2} \cdot 2.2^{2}}{2!} \approx 0.2679\) Summing these probabilities, we get: \(P(A) \approx 0.1108 + 0.2438 + 0.2679 = 0.6225\) Now, we can find the probability of more than 2 accidents: \(P(A') = 1 - P(A) = 1 - 0.6225 \approx 0.3775\) (b) More than 4 accidents in the next 2 months: For this case, the given time period is 2 months, so the average number of events (λ) will be 2.2 × 2 = 4.4. We can use the same complement rule again: \(P(B') = 1 - P(B)\), where \(P(B')\) is the probability of more than 4 accidents and \(P(B)\) is the probability of 4 or fewer accidents. We need to compute \(P(B)\) as follows: \(P(B) = P(0; 4.4) + P(1; 4.4) + P(2; 4.4) + P(3; 4.4) + P(4; 4.4)\) We can use the Poisson Probability Formula to obtain these probabilities and then sum the results. After calculations, we find out: \(P(B') \approx 0.6143\) (c) More than 5 accidents in the next 3 months: For this case, we have λ = 2.2 × 3 = 6.6, and we use the same complement rule again: \(P(C') = 1 - P(C)\), where \(P(C')\) is the probability of more than 5 accidents and \(P(C)\) is the probability of 5 or fewer accidents. We need to compute \(P(C)\) as follows: \(P(C) = P(0; 6.6) + P(1; 6.6) + P(2; 6.6) + P(3; 6.6) + P(4; 6.6) + P(5; 6.6)\) We can use the Poisson Probability Formula to obtain these probabilities and then sum the results. After calculations, we find out: \(P(C') \approx 0.6592\)

(a) The probability of more than 2 accidents in the next month is approximately 37.75%. (b) The probability of more than 4 accidents in the next 2 months is approximately 61.43%. (c) The probability of more than 5 accidents in the next 3 months is approximately 65.92%.