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 on a certain section of highway during a two-week period, we can use the Poisson distribution formula.

Given:
- Average number of traffic accidents per week = 2
- Time period = 2 weeks

First, we need to calculate the average number of accidents for the two-week period:
$$ \text{Average accidents in 2 weeks} = 2 \times 2 = 4 $$

Now, we can use the Poisson distribution formula to find the probability of 3 accidents during the two-week period:
$$ P(X = 3) = \frac{e^{-\lambda} \cdot \lambda^3}{3!} $$

where:
- $$ \lambda $$ is the average number of accidents in the given time period (4 in this case)
- $$ e $$ is the base of the natural logarithm (approximately 2.71828)
- $$ 3! $$ is the factorial of 3 (3 * 2 * 1 = 6)

Substitute the values into the formula:
$$ P(X = 3) = \frac{e^{-4} \cdot 4^3}{3!} $$

Now, we can calculate the probability using the formula.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{e^{-4}\times 4^{3}}{3!}$$
- step1: Multiply the numbers:
  $$\frac{\frac{4^{3}}{e^{4}}}{3!}$$
- step2: Multiply by the reciprocal:
  $$\frac{4^{3}}{e^{4}}\times \frac{1}{3!}$$
- step3: Multiply the terms:
  $$\frac{4^{3}}{e^{4}\times 3!}$$
- step4: Multiply the terms:
  $$\frac{4^{3}}{6e^{4}}$$
- step5: Factor the expression:
  $$\frac{4^{3}}{2\times 3e^{4}}$$
- step6: Reduce the fraction:
  $$\frac{2^{5}}{3e^{4}}$$
- step7: Evaluate the power:
  $$\frac{32}{3e^{4}}$$
The probability of 3 accidents on this section of highway during a two-week period is approximately 19.5%.
The probability of 3 accidents during a two-week period is approximately 19.5%.

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)
b)
c)
d)

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