The VIP cafeteria door on the Death Star promptly opens at 11:00 am and closes at 1:00 pm (Standard Galactic Time). Nobody is allowed to enter at other times but guests can stay until they finish their meal. To keep their lean physiques, Sith Lords usually spend their allotted 14-minute lunch break in the cafeteria sipping organic kale smoothies. Darth Sidious has a yoga class at 11:00 am, so he never has lunch before noon. Darth Vader must use a straw, so he is allowed an additional 4 minutes to slurp his smoothie. What is the probability that the two of them meet today in the cafeteria?
Here's what I have so far. I know that, in order for Darth Vader and Darth Sidious to meet, Darth Vader must enter the cafeteria no earlier than 11:42. I also know that Darth Sidious has to enter between 12:00 and 1:00. Because the guests are allowed time to finish their meals before getting kicked out of the cafeteria, I know that Darth Vader and Darth Sidious could both enter around 1 pm, and stay in the cafeteria together until no later than 1:14.
I just don't know how to structure my work. It might be easier to entertain the case where they both eat for the same duration of time of 14 mins, and then after computing that, see what adjustments have to made so that we can find the probability when Darth Vader takes 18 mins. Any help is massively appreciated!