9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number \(X\) of these discs that are black.
- Find \(\mathrm { P } ( X = 3 )\).
Paula decides to use the binomial distribution as a model for the distribution of \(X\).
- Explain why this model will give probabilities that are approximately, but not exactly, correct.
- Paula uses the binomial model to find an approximate value for \(\mathrm { P } ( X = 3 )\). Calculate the percentage by which her answer will differ from the answer in part (ii).
Paula now assumes that the binomial distribution is a good model for \(X\). She uses a computer simulation to generate 1000 values of \(X\). The number of times that \(X = 3\) occurs is denoted by \(Y\).
- Calculate estimates of the limits between which two thirds of the values of \(Y\) will lie.