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Two fair six-sided dice are rolled and observe the two outcomes. Let the random variables Y and Z denote the outcomes observed on the two dice and let X = Y + Z. Find the probabilities P(X = x), P(Y = y), P(Z = z) and the conditional probability P(Y = y | X = x) where x, y and z are the values that random variables X, Y and Z take respectively in their range. Compare the calculated probabilities with the probabilities obtained by Monte Carlo Simulation.
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