Given a K*N grid with K > N. fill each row with only one marble at random position. What is the expected number of empty columns?
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With regards to the edit, finding the expected number of empty columns, this is easiest solved using linearity of expectation. Find the probability the first column was empty. That should be simple since it is merely the event where every marble was placed in a column different than the first. Use that value you calculated and multiply it by the number of columns to get the expected number. – JMoravitz Oct 06 '21 at 18:22
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The probability would be (N-1/N)^K for none filled 1st column or any column? There are also cases that more than one column is not filled (2,3..N-1). Would multiplying by N count those scenarios as well? – Snownorth Oct 07 '21 at 00:33
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That is the point, yes. Linearity of expectation is very powerful and does not require that events be independent or otherwise... $E[X+Y]=E[X]+E[Y]$ always. Here, letting $X_1$ be the event that the $1$'st column is empty, similarly $X_2,X_3,\dots$ we have the total number of empty columns is $X_1+X_2+\dots+X_N$ and the expected number is $E[X_1+X_2+\dots+X_n]=E[X_1]+E[X_2]+\dots+E[X_N]=N\cdot E[X_1]$ due to symmetry. – JMoravitz Oct 07 '21 at 03:30