An urn contains 100 balls that have the numbers 1 to 100 painted on them (every...

An urn contains 100 balls that have the numbers 1 to 100 painted on them (every ball has a distinct number). You keep sampling balls uniformly at random (i.e., every ball is equally likely to be picked), one at a time, and without replacement. For 1 ? i < j ? 100, let E_{i,j} denote the event that the ball with the number j was taken out of the urn before the ball with the number i. Prove that the events E_{45,89} and E_{23,60} are independent. Are E_{13,72} and E_{72,99} also independent? Why or why not?

Hint: You might want to think of the outcomes as permutations of 1 to 100, and ? as the set of all possible permutations of 1 to 100 (why?).

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orchestra answered 2 years ago

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