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.
For an event with n competitors there will then be a sequence of announced ranks. What is the expected average of these ranks?
In long track speed skating, 2n competitors go on the same track but in pairs. So the first pair will get announced ranks of 1 and 2, the next pair will have two ranks from {1,2,3,4}, and so on.
For an event with n pairs, what's the expected average of the ranks?
(In reply to
re: solution by tomarken)
On another reading of the problem, your interpretation does sound more plausible, with only one new announced tank after every competitor has his/her turn.
I think there's an intermediate interpretation. For example, if the second competitor is better than the first then of course the second competitor is announced as the new number 1. But at the same time the previous #1 then becomes #2.
Under my first interpretation there are always k announced ranks after the kth performance. Under your interpretation there is only one announced rank after each performance. Under the intermediate there can be anywhere from 1 to k announced (possibly revised) ranks, depending on where the current performer falls among the previous performers.
But it sounds as if your interpretation is the correct one.
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Posted by Charlie
on 2022-02-16 23:18:27 |
re(2): solution
