Magic Kingdom park is a theme park at Walt Disney World Resort featuring classic attractions, enchanting fireworks, musical parades and Disney Characters. But while theme park guests enjoy the various attractions, engineers are hard at work in an underground control room making decisions designed to improve the guest experience for all. Forecasting serves as the analytical foundation for operations planning at the Resort. It all starts with the park attendance forecast, which lays out the expected attendance at each park. These predictions are strongly considered when setting park hours and performing other strategic planning. Another innovative way the resort uses forecasting is for managing and estimating attraction wait times. The most popular attractions utilize Disney's FastPass system - a unique virtual queueing system that allows guests to receive a ticket with a designated one-hour window of time when they can return and skip the regular (standby) line. The virtual queue poses an extra challenge to forecasting the standby wait time since many FastPass guests will return before the guest entering the standby queue reaches the front of the line. For a single theme park per day, you can make up to 3 FastPass selections in advance. The FastPass system poses a challenge to forecasting because wait time estimates are negatively affected as the percentage of individuals who miss their one-hour reservation window increases. (This happens more frequently for the third FastPass because after each park guests uses their allotted three FastPasses, they receive additional FastPasses, one at a time, for any Walt Disney World park until closing or all passes have been reserved, whichever comes first. To book this "fourth" FastPass, guests must have three booked at the first park; thus, if they only want to ride two attractions at the first park, they book a third and simply do not board the ride.) Additional information regarding Magic Kingdom follows:

Magic Kingdom has a total of 42 attractions, 25 of which offer FastPass tickets12% of the guests in line for an attraction that offers FastPass tickets have a FastPass reservation, the others wait in the regular (standby) line7% of park visitors are Florida residents10% of park visitors are annual pass holdersThe arrival of guests in the line at Splash Mountain can be modeled as a Poisson process; the variance of the number of guests that arrive in a 10-minute period is 12 guests.

1. If three attractions breakdown, what is the probability that two of them are attractions for which FastPass tickets are offered?

2. Historical data shows that annual passholders are less likely to miss their FastPass reservations. If a current FastPass queue for the period from 1pm to 2pm at Splash Mountain has 50 guests, what is the expected number of annual passholders among these 50 guests?

The probability that a FastPass reservation will be missed given that it is the first, second, or third scheduled is 3%, 5%, and 18%, respectively. Not all FastPasses are used, thus, the probability that any given FastPass is the first one scheduled is 37%. The probability that a FastPass is the second one scheduled is 35%. The remaining 28% of FastPasses scheduled are the third scheduled.

3. If a FastPass reservation was missed, what is the probability that it was the third FastPass used?

4. What is the probability that eight guests will arrive at Splash Mountain in a 5-minute period?