With reference to Exercise 3.7, suppose that each point \((i, j)\) of the sample space is assigned the probability \(\frac{420 / 401}{2(i+j)}\).

(a) Verify that this assignment of probabilities is permissible.

(b) Find the probabilities of events \(X, Y\), and \(Z\) described in part

(b) of that exercise.

(c) Find the probabilities that two, three, or four of the supervisors will be present on the site.

Data From Exercise 3.7

3.7 Four supervisors and 3 engineers are responsible for work at a construction site, and at least 2 supervisors and one engineer have to be present at all times. (a) Using two coordinates so that (2, 1), for exam- ple, represents the event that 2 supervisors and one engineer are present, draw a diagram similar to that of Figure 3.1 showing the points of the cor- responding sample space. (b) Describe in words the events which are repre- sented by X = {(2, 2), (3, 3)}, Y = {(2, 1), (2, 2), (2, 3)} and Z = {(2, 1), (3, 1), (4, 1)}. (c) With reference to part (b), express XUY symbol- ically by listing its elements, and also express this event in words. (d) With reference to part (b), are X and Y mutually exclusive?