A company manager has two secretaries, Mary and Joanne. In each page typed by Mary, the number of misprints can be 0, 1, or 2 with probabilities 0.50, 0.35, and 0.15, respectively, while the number of misprints in a page typed by Joanne can also be 0, 1, or 2, but with probabilities 0.60, 0.30, and 0.10, respectively. The manager uses Mary more often to type his documents, and in fact she types 50%

more pages than Joanne does.

Among the pages typed, we select one at random, without knowing who typed it.

Let X be the number of misprints on that page.

(i) Use the law of the total probability (Proposition 3.4) to calculate the probabilities P(X = 0), P(X = 1), P(X ≥ 1).

(ii) Obtain the distribution function of X.