Dice. When two balanced dice are rolled, 36 equally likely outcomes are possible, as illustrated in Fig. 4.1 on page 181. Let A = event the red die comes up even, B = event the black die comes up even, C = event the sum of the dice is even, D = event the red die comes up 1, 2, or 3, E = event the red die comes up 3, 4, or 5, and F = event the sum of the dice is 5.

Apply the definition of independence for three events stated in Exercise 4.204 to solve each problem.

a. Are A, B, and C independent events?

b. Show that P(D & E & F) = P(D) · P(E) · P(F) but that D, E, and F are not independent events.