Question

Please help with this Statistical Analysis problem. Thank you so much!

Railway track allocation. Refer to the Journal of Trans- portation Engineering (May, 2013) investigation of the as- signment of tracks to trains at a busy railroad station, Exercise 2.8 (p. 28). Ideally, engineers will assign trains to tracks in order to minimize waiting time and bottlenecks. Assume there are 10 tracks at the railroad station and the trains will be randomly assigned to a track. Suppose that in a single day there are 50 trains that require track assignment. a. What is the probability that exactly 5 trains are as- signed to each of the 10 tracks? b. A track is considered underutilized if fewer than 2 trains are assigned to the track during the day. Find the probability that Track #1 is underutilized.

Answer 1

Given that, total track = 10 total trains = 50 (a): We need to calculate probability that exactly 5 train is assigned to each 10 track. Total no. of ways of assign trains to any track = 10^50 (Since Each train any tracks 10, and there are 10 trains). Total no. of ways 5 trains is assigned to each 10 track. = 50 ! (5 ! times 5 ! ... 10 times) = 50 !/(5 !)^10 probability that exactly 5 train is assigned to each 10 track = (50 !/5 !)^10)/10^50) = 4.91204585 times 10^- 7 \r\n b): Number of ways that fewer than 2 trains are assigned to track 1 = (Number of ways that 0 trains are assigned to track 1 + Number of ways that 1 train are assigned to track 1) Number of ways that 0 trains are assigned to the track 1 = 9^50 (Since there are 50 trains, and 9 tracks). Number of ways that 1 train are assigned to track 1 = 50 times 9^49