462) The average number of traffic accidents on a certain section of highway is two per week. Assuming that the number of accidents follow a poisson distribution. Find the probability of 3 accidents on this section of highway during two-week period. a) \( \mathbf{1 9 . 5 \%} \) b) \( 28.9 \% \) c) \( 23.8 \% \) d) \( 12.8 \% \)
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To find the probability of 3 accidents occurring in a two-week period, we first determine the average rate (λ) for that time frame. Since the average rate is 2 accidents per week, over two weeks, the average (λ) is \( 2 \times 2 = 4 \). Using the Poisson probability formula: \[ P(X=k) = \frac{λ^k e^{-λ}}{k!} \] For \( k = 3 \) and \( λ = 4 \): \[ P(X=3) = \frac{4^3 e^{-4}}{3!} \] \[ = \frac{64 e^{-4}}{6} \] \[ \approx \frac{64 \times 0.0183}{6} \] \[ \approx 0.1953 \, (or \, 19.53\%) \] Thus, the answer is **a) 19.5%**.