Skylines – Racing against the clock SkyWings Airlines is one of Asia’s fastest-growing carriers, known for its strict on-time departure policy. To maintain punctuality, the airline closes boarding gates 10 minutes before departure without exception. At Metro International Airport, where SkyWings operates over 120 flights daily, the passenger experience is often shaped by the unpredictable wait at the security check. According to airport statistics, the average waiting time is 20 minutes, and the distribution of wait times follows an exponential pattern. Some passengers clear security in under 5 minutes, while others have waited nearly an hour during peak times. On a Monday morning, Ms. Thennal K. Warrier, a frequent flyer on SkyWings, arrived at the airport for her Flight 503 to Bengaluru. She reached the security checkpoint at 8:20 AM, exactly 40 minutes before the 9:00 AM scheduled departure. If her security wait exceeds 25 minutes, she will miss her flight, despite arriving at the airport 40 minutes early. Her situation unfolded as follows: 8:20 AM – Arrival at security: Ms. Warrier joins the line. Time buffer available: o Gate closes at 8:50 AM (10 minutes before departure). o Walking time from security to gate: 5 minutes. o Effective cutoff time for clearing security: 8:45 AM. Total time available at security: 25 minutes (from 8:20 AM to 8:45 AM). Airport records reveal troubling patterns: On average, 18% of passengers wait more than 40 minutes in the queue. Even after waiting for 20 minutes, many passengers often wait an additional 20 minutes or more due to congestion. Surveys show that 62% of travelers check their phones for updates within 15 minutes of entering security, highlighting growing anxiety about missing flights. SkyWings has observed that such delays directly impact customer satisfaction. In the last quarter alone, 6% of passengers on morning flights missed boarding because of long security lines, despite being present in the terminal on time. The airline is now considering three options: Passenger Alerts – Send reminders to arrive at least 1.5–2 hours before departure for busy airports. Fast-Track Lanes – Partner with airport authorities to offer express lanes for premium customers. Predictive Updates – Use analytics to send real-time security wait estimates to passengers via the SkyWings mobile app. Questions 1. What is the probability that Ms. Warrier will miss her flight? 2. What is the probability that she will wait more than 40 minutes at security? 3. If she has already waited 20 minutes, what is the probability that she will need to wait another 20 minutes? 4. To ensure a 99% chance of catching her flight, how early should Ms. Warrier reach the security checkpoint (in hours before departure)?