Question: Suppose that 60% of the supply of raw material kits used in a chemical reaction can be classified as recent, 30% as moderately aged, 8% as aged, and 2% unusable. Sixteen kits are randomly chosen to be used for 16 chemical reactions. Let N1, N2, N3, N4 denote the number of chemical reactions performed with recent, moderately aged, aged, and unusable materials.
(a) Find the probability that exactly one of the 16 planned chemical reactions will not be performed due to unusable raw materials.
(b) Find the probability that 10 chemical reactions will be performed with recent materials, 4 with moderately aged materials, and 2 with aged materials.
(c) Use an R command to recalculate the probabilities in part (b).
(d) Find Cov(N1 + N2, N3) and explain, at an intuitive level, why it is reasonable for the covariance to be negative.
(e) Find the variance of N1 + N2 + N3. (Hint. Think of N1 + N2 + N3 as binomial.)