You have an upcoming trip to Washington D.C. and you are fascinated with the intricacies of Congressional committee hearings. You wish to attend as many hearings as possible during your trip, and your local representative has provided you with a pass that will get you into the audience of any hearing. But there are some challenges in planning your schedule.
Specifically:
Your goal is to develop a strategy that maximizes the expected number of hearings that you can attend during your trip. As an example, consider a situation in which there are four hearings with parameters as follows:
hearing  s  a  b 

Social media and elections 



NASA missions 



Oil and gas exploration 



Hurricane recovery efforts 



For this schedule, the optimal strategy will allow you to achieve an expected value of 2.125 hearings. To achieve this, you begin by attending the NASA hearing, which starts at time 3 and ends with equal probability at either time 5 or time 6 (given the hearing length that is uniformly distributed over {2, 3}). If the NASA hearing does end at time 5 you will immediately head to the oil and gas exploration hearing, and there is a 14 chance that hearing will end at time 6, allowing you to make yet a third hearing (about hurricane recovery efforts). If the NASA hearing instead ends at time 6, you will go straight to the hurricane hearing.
By this strategy you will attend 3 hearings 12.5% of the time and two hearings the other 87.5% of the time, and thus expected value of 2.125. Note that if you were to start by attending the social media and elections hearing, you might optimistically make four hearings. However, a careful analysis will demonstrate that if you attend the first hearing, your optimal expected value is only 2.10714.
For each case, display the expected number of hearings of an optimal strategy. Your answer should have an absolute or relative error of at most 10^{−6}.
2 4 1 1 7 3 2 3 5 1 4 6 10 10 5 1 1 7 1 1 6 3 2 3 5 1 4 6 10 10
2.125000 2.291667