https://file.aiquickdraw.com/mathbot/file/1729619569166rxhf5xni.jpg

Let's analyze the image step by step.

### Part (a): What does 17.8 represent in the probability model?
The probability model shown is for a Poisson distribution, which models the number of events occurring in a fixed interval of time. In a Poisson distribution, the parameter (λ) represents the average number of events in the interval. In this case, 17.8 represents the **average number of people** entering the security check-in lineup every 15 minutes. This corresponds to **Option B**.

### Part (b): Compute the probability that 15 people enter the security check-in lineup in a 15-minute interval.
The Poisson distribution probability formula is:
\[
P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}
\]
Where \(\lambda = 17.8\) and \(x = 15\). The answer already given is correct: \(P(X = 15) = 0.0812\).

### Part (c): Compute the probability that at least 4 people will enter the security check-in lineup in a 5-minute interval.
For this, you need to compute the cumulative probability of fewer than 4 people entering and subtract it from 1. The answer given, 0.8427, is correct for the specified interval.

### Part (d): In the past 15 minutes, you have been told that somewhere between 15 and 20 people, inclusive, have entered the security lineup. Compute the probability that this uncertain number is 17.
To solve this, we calculate the probability for each value from 15 to 20 and then divide the probability for \(X = 17\) by the sum of the probabilities from 15 to 20. The formula used will be:

\[
P(X = 17 \mid 15 \leq X \leq 20) = \frac{P(X = 17)}{P(15 \leq X \leq 20)}
\]

This involves calculating each individual probability for \(X = 15, 16, 17, 18, 19, 20\), summing them up, and then dividing the value for \(X = 17\) by this sum.

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Would you like more details on how to compute part (d) explicitly?

### 5 Related Questions:
1. How is the Poisson distribution different from other distributions like the binomial or normal distribution?
2. Why is the Poisson distribution used for modeling events like this?
3. Can we apply the same model for different time intervals (e.g., 10 minutes instead of 15)?
4. What are some real-world applications of the Poisson distribution beyond airport lineups?
5. How does changing the average rate (λ) affect the probabilities in a Poisson distribution?

### Tip:
For Poisson problems, always ensure you're working with the correct time interval and average event rate for accurate results.

The number of people entering a security check-in lineup in a 15-minute interval at a medium-sized airport can be modeled by a Poisson distribution with a mean (λ) of 17.8. Compute the following:
(a) What does 17.8 represent in the probability model?
(b) Find the probability that 15 people enter the security check-in lineup in 15 minutes.
(c) Find the probability that at least 4 people enter the lineup in 5 minutes.
(d) Given that between 15 and 20 people (inclusive) have entered in the past 15 minutes, find the probability that exactly 17 people entered.

Solve a Poisson distribution problem involving airport security check-ins with an average rate of 17.8 per 15 minutes. Learn to compute probabilities for specific events, including exact counts and conditional probabilities. Suitable for Grades 11-12.

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