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Announced vs Final rank average (Posted on 2022-02-23) Difficulty: 3 of 5
In some televised sports, such as downhill skiing, individual competitors take turns for the best time on a course. After each competitor the announcers will give the current standing. So for example, the first person will always be announced as ranked 1 (though this will likely change), the second person will be announced as either 1 or 2, and so on.

By the end of the competition everyone's final rank will be the same or higher.

a) In an event with n competitors, what is the expected difference between the first announced and final rank of competitor that goes xth?

b) What is the expected average difference for all n competitors?

Note: the order of the competitors is random.

No Solution Yet Submitted by Jer    
Rating: 5.0000 (1 votes)

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Solution solution | Comment 1 of 2
a) The final rank of course will be x. The initial rank depends on how many of those with lower rank precede him. There are x-1 whose ultimate rank are lower out of the n-1 competitors to person x.

For example, if x = 3, there are 2 competitors with lower numbered rank than him. Both can appear before him, one before and one after, or both after. Each has probability 1/3. If both appear befor him, the required difference = 0; one before and one after, the difference is 1; if both after, the difference is 2.

x  prob  diff
1   1      0
2   1/2    1
    1/2    0
    
3   1/3    0
    1/3    1
    1/3    2

...

In all instances there are equal probabilities of starting zero, 1, 2, etc up to x-1 positions away and the average is (x-1)/2.

To see this better, remember we can ignore the placement in time of those with higher rank number. Among those with rank x or a lower number, person X has an equal likelihood of being the first, second, ..., xth.

b) Example for n=5:

   0
   1/2
   1
   3/2
   2
   
 The average is 1 or half of (n-1)/2, i.e. (n-1)/4.  

  Posted by Charlie on 2022-02-23 10:14:42
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