3. Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 10 passengers per minute. Compute the probability of at least one arrival in a 15-second period.

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**Problem Statement:**

Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 10 passengers per minute. Compute the probability of at least one arrival in a 15-second period.

**Solution Outline:**

To tackle this problem, we can use the Poisson distribution, which is generally used for modeling the number of times an event happens in a fixed interval of time or space.

1. **Convert the Time Unit:**
   - Mean arrival rate per minute: \( \lambda = 10 \) passengers.
   - Convert the rate to a 15-second period.
   Since there are 60 seconds in a minute, the rate per 15 seconds is:
   \[
   \lambda_{15s} = \frac{10 \text{ passengers}}{60 \text{ seconds}} \times 15 \text{ seconds} = 2.5 \text{ passengers}
   \]

2. **Poisson Probability Formula:**
   The Poisson probability of observing \( k \) events in a given interval is:
   \[
   P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}
   \]
   where \( \lambda \) is the average number of events (in this case, 2.5), and \( k \) is the actual number of events.

3. **Probability of at Least One Arrival:**
   The probability of at least one arrival is 1 minus the probability of no arrivals.
   - Probability of no arrivals (k=0):
   \[
   P(X = 0) = \frac{e^{-2.5} \cdot 2.5^0}{0!} = e^{-2.5}
   \]
   - Therefore, the probability of at least one arrival:
   \[
   P(X \geq 1) = 1 - P(X = 0) = 1 - e^{-2.5}
   \]

4. **Compute the Exponential:**
   Using the value of \( e^{-2.5} \approx 0.0821 \):
   \[
   P(X \geq 1) = 1 - 0.0821 = 0.9179
   \]

**Conclusion:**
The probability of at least one passenger arriving in a 15-second period at the passenger
Transcribed Image Text:**Problem Statement:** Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 10 passengers per minute. Compute the probability of at least one arrival in a 15-second period. **Solution Outline:** To tackle this problem, we can use the Poisson distribution, which is generally used for modeling the number of times an event happens in a fixed interval of time or space. 1. **Convert the Time Unit:** - Mean arrival rate per minute: \( \lambda = 10 \) passengers. - Convert the rate to a 15-second period. Since there are 60 seconds in a minute, the rate per 15 seconds is: \[ \lambda_{15s} = \frac{10 \text{ passengers}}{60 \text{ seconds}} \times 15 \text{ seconds} = 2.5 \text{ passengers} \] 2. **Poisson Probability Formula:** The Poisson probability of observing \( k \) events in a given interval is: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] where \( \lambda \) is the average number of events (in this case, 2.5), and \( k \) is the actual number of events. 3. **Probability of at Least One Arrival:** The probability of at least one arrival is 1 minus the probability of no arrivals. - Probability of no arrivals (k=0): \[ P(X = 0) = \frac{e^{-2.5} \cdot 2.5^0}{0!} = e^{-2.5} \] - Therefore, the probability of at least one arrival: \[ P(X \geq 1) = 1 - P(X = 0) = 1 - e^{-2.5} \] 4. **Compute the Exponential:** Using the value of \( e^{-2.5} \approx 0.0821 \): \[ P(X \geq 1) = 1 - 0.0821 = 0.9179 \] **Conclusion:** The probability of at least one passenger arriving in a 15-second period at the passenger
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