Please Show Detail. I don\'t get part B. need to why and how. 4.46 Railway track
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Please Show Detail. I don't get part B. need to why and how.
4.46 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.Explanation / Answer
1) total track = 10
total trains = 50
a) we need to calculate probability that exactly 5 train is assigned to each 10 track ,
total number of ways of assigng trains to any track = 10 ^50
{since each train can go to any of 10 tracks,and there are 10 trains}
total number of ways 5 train is assigned to each 10 track
= 50!(5!*5!... 10 times) = 50 ! /(5!)10
probability that exactly 5 train is assigned to each 10 track , = (50 ! /(5!)10) / (10 ^50 ) = 4.91204585 *10-7
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 track 1 = 9^50 { since there are 50 trains , and 9 tracks }
number of ways that 1 train are assigned to track 1 = 50*9^49 { there are 50 ways to choose 1 train and rest 49 trains can assigned to 9 tracks by 9^49 ways}
= 950 + 50*949
hence probability = (950 + 50*949/10^50)=0.03378585969
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