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Conditional expectation of discrete random variables: Practice

Problem 1. Let \(X\) be the number of rolls of a fair die until the obtained number is \(6\). Let \(Y\) be the integer drawn uniformly at random from the set \(\{1,2,\dots, X\}\). Calculate \(\mathbb E\left[\left.Y\right|X\right]\).

Problem 2. Use the result from problem 1 to calculate \(\mathbb E\left[Y\right]\). The random variable \(Y\) is defined in problem 1.

Problem 3. A die is rolled until the number \(6\) is obtained. What is the expected number of throws (including the throw giving \(6\)) conditioned on the event that all throws gave even numbers?

Problem 4. Starting with an empty \(1\times n\) board (a row of \(n\) squares), we successively place \(1\times 2\) dominoes to cover two adjacent squares. At each stage, the placement of the new domino is chosen at random, with all available pairs of adjacent empty squares being equally likely. The process continues until no further dominoes can be placed. Find the limit, as \(n\) goes to infinity, of the expected fraction of the board that is covered when the process ends.