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An urn contains \(N\) balls (\(N>50\)), of which \(50\) are black and the others are white. Ryem performs \(n\) draws with replacement and records the proportion of white balls obtained.
\(n\) (Number of trials) 100 1,000 10,000
Proportion of white balls 0.753 0.748 0.7499
  1. Define a random variable \(X_i\) associated with the \(i\)-th draw (value \(1\) for a white ball and \(0\) for a black ball). What is the distribution of \(X_i\)? Give its parameter \(p\) in terms of \(N\).
  2. Express the proportion of white balls observed after \(n\) draws as a sample mean \(\overline{X}_n\).
  3. Using the Law of Large Numbers and the result for \(n=10\,000\), give an estimate of \(p\).
  4. Deduce an estimate of \(N\) by solving the equation linking \(p\) and \(N\).

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