With reference to Exercise 3.1, suppose that the points \((0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3)\), \((2,0),(2,1),(2,2),(2,3),(3,0),(3,1),(3,2)\), and \((3,3)\) have the probabilities \(0.080,0.032,0.086,0.064\), \(0.085,0.073,0.065,0.091,0.071,0.050,0.046,0.075\), \(0.040,0.021,0.080\), and 0.041.

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

(b) Find the probabilities of events \(A, B\), and \(C\) given in part

(b) of that exercise.

(c) Calculate the probabilities that one, two, or three supplies of white cement are adulterated.

Data From Exercise 3.1

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3.1 A civil engineer suspects that the cement being sup- plied for constructing a dam is adulterated. There are three units of white cement and three units of black cement. He will check all six for adulteration. (a) Express each outcome using two coordinates, so (1, 2), for example, represents the event that one of the white cement units and two of the black cement units are adulterated. Draw a dia- gram similar to that of Figure 3.1 showing the 16 outcomes in the sample space. (b) If A is the event that equally many white cement and black cement units are adulterated, B is the event that none of the white cement units is adul- terated, and C is the event that fewer white cement units are adulterated than black cement units, ex- press each of these events symbolically by listing its elements.