This problem involves a Poisson distribution, where the mean number of accidents is 2 per week. I'll break it down step by step to calculate the probabilities.

Definitions:

For a Poisson random variable $X$ with mean $\lambda$: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$

Scaling:

• The mean is given as $2$ accidents per week.
• For $3$ weeks, $\lambda = 3 \times 2 = 6$.
• For $4$ weeks, $\lambda = 4 \times 2 = 8$.

Calculations:

a) Probability of no accidents in 3 weeks:

$P(X = 0) = \frac{\lambda^0 e^{-\lambda}}{0!} = e^{-\lambda}$ Substitute $\lambda = 6$: $P(X = 0) = e^{-6}$

b) Probability of less than 2 accidents in 4 weeks:

"Less than 2 accidents" means $P(X < 2) = P(X = 0) + P(X = 1)$.

For $\lambda = 8$: $P(X = 0) = e^{-8}$ $P(X = 1) = \frac{8^1 e^{-8}}{1!} = 8 \cdot e^{-8}$ $P(X < 2) = P(X = 0) + P(X = 1) = e^{-8} + 8 \cdot e^{-8} = 9 \cdot e^{-8}$

c) Probability of more than 1 accident in 4 weeks:

"More than 1 accident" means $P(X > 1) = 1 - P(X \leq 1)$.

We already calculated $P(X \leq 1)$ in part (b): $P(X \leq 1) = P(X = 0) + P(X = 1) = 9 \cdot e^{-8}$ So: $P(X > 1) = 1 - P(X \leq 1) = 1 - 9 \cdot e^{-8}$

Let me compute these probabilities to 10 decimal places. math

Part a: Probability of no accidents in 3 weeks (λ = 6)

lambda_3_weeks = 6 P_no_accidents_3_weeks = math.exp(-lambda_3_weeks)

Part b: Probability of less than 2 accidents in 4 weeks (λ = 8)

lambda_4_weeks = 8 P_0_accidents_4_weeks = math.exp(-lambda_4_weeks) P_1_accident_4_weeks = (lambda_4_weeks ** 1) * math.exp(-lambda_4_weeks) / math.factorial(1) P_less_than_2_4_weeks = P_0_accidents_4_weeks + P_1_accident_4_weeks

Part c: Probability of more than 1 accident in 4 weeks

P_more_than_1_4_weeks = 1 - P_less_than_2_4_weeks

Format results to 10 decimal places

results = { "a": round(P_no_accidents_3_weeks, 10), "b": round(P_less_than_2_4_weeks, 10), "c": round(P_more_than_1_4_weeks, 10) }

resultsHere are the probabilities calculated to 10 decimal places:

• a) Probability of no accidents in 3 weeks: 0.0024787522
• b) Probability of less than 2 accidents in 4 weeks: 0.0030191637
• c) Probability of more than 1 accident in 4 weeks: 0.9969808363

Let me know if you need further explanations or additional computations.

Related Questions:

1. What is the general formula for the Poisson probability distribution?
2. How does the parameter $\lambda$ affect the shape of the Poisson distribution?
3. Could we calculate similar probabilities for other intervals (e.g., 5 weeks)?
4. What is the expected number of accidents in 4 weeks?
5. How can we verify these probabilities using statistical software or tools?

Tip:

In Poisson distributions, probabilities drop off rapidly as $k$ moves away from the mean $\lambda$. This property can simplify approximations for extreme cases.