I saw a solution for Little Pony and Expected Maximum but couldn't really understand why it works. Can someone explain the logic behind it?
I have seen the tutorial, but it doesn't explain this approach.
double p = m;
for(int i = 1; i < m; i++)
p -= pow((double) i / m, n);
printf("%.16f\n", p);
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You have troubles in simplifying sum from tutorial to this one? I written solution but can't insert image for some reason =/ So, here.
Oh. Thank you. I had thought that this was a different, perhaps cleverer, approach. Didn't think it was just a reduced form.
Well, but problem remains the same, right? :) Almost in any math problem you can prove some equivalence between different approaches. Maybe, there is also some naive explanation of this answer... But I don't know. At that contest I had terrible formula (7314095), but it can be reduced to this one too.
The derivation of the formula in tutorial is quite simple —
Let us assume that we have Xj which denotes the number of throws when the maximum number appearing on dice is j, So Xj is given by —
Xj = jn — (j-1)n
(Assume that we have n places to fill such that their maximum is j, So the arrangement should contain at least one j, so consider each permutation formed by numbers from 1 to j and subtract each permutation formed by numbers from 1 to (j-1). The resulting value will contain each permutation such that it contains at least one j)
Now finding out the expected value is quite easy
Hope it helps :)