Compute the probability that between 1 and 9 customers will enter the convenience store within the next 10-minutes?
P(1 < X < 9)
0.9315

To calculate the probability that between 1 and 9 customers will enter the convenience store within the next 10 minutes, given \( P(1 < X < 9) = 0.9315 \), we can assume that this is a Poisson distribution problem (common for counting events over time).

The Poisson probability mass function is given by:

\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

Where:
- \( \lambda \) is the average rate of occurrences (in this case, the average number of customers per 10 minutes),
- \( k \) is the number of occurrences.

Since you have the probability that between 1 and 9 customers enter the store as \( P(1 < X < 9) = 0.9315 \), the exact details about \( \lambda \) (the rate parameter) are not provided explicitly. But we can infer that the calculation involves finding:

\[
P(2 \leq X \leq 8) = P(X < 9) - P(X \leq 1)
\]

Where:
- \( P(X < 9) \) gives the cumulative probability of fewer than 9 customers,
- \( P(X \leq 1) \) gives the probability of 0 or 1 customer entering.

Since the final probability is already provided as 0.9315, these cumulative values have been calculated beforehand.

### Would you like more details on solving Poisson distribution problems, or any specific part of the steps above?

Here are 5 relative questions to expand the problem:
1. What is the rate \( \lambda \) of customer arrivals per minute?
2. How would you calculate \( P(X = 2) \) in this case?
3. How do you compute \( P(X = 0) \) or \( P(X = 1) \) for different values of \( \lambda \)?
4. How does changing \( \lambda \) affect the distribution of customers?
5. What is the cumulative distribution function (CDF) for the Poisson distribution?

**Tip:** The Poisson distribution is ideal for modeling rare events over a fixed period of time or space!

Compute the probability that between 1 and 9 customers will enter the convenience store within the next 10-minutes? P(1 < X < 9)

Learn how to calculate the probability that between 1 and 9 customers will enter a convenience store within 10 minutes using the Poisson distribution. This solution covers essential probability concepts, the Poisson probability mass function, and how to compute cumulative probabilities. Suitable for advanced high school or college-level students.

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