# Independence

Just as variables and observations can be independent, random processes can be independent, too. Two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other. For instance, flipping a coin and rolling a die are two independent processes – knowing the coin was heads does not help determine the outcome of a die roll. On the other hand, stock prices usually move up or down together, so they are not independent.

Example  provides a basic example of two independent processes: rolling two dice. We want to determine the probability that both will be 1. Suppose one of the dice is red and the other white. If the outcome of the red die is a 1, it provides no information about the outcome of the white die. We first encountered this same question in Example, where we calculated the probability using the following reasoning: 1/6 of the time the red die is a 1, and 1/6 of those times the white die

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15(a) The complement of A: when the total is equal to 12. (b) P (Ac) = 1/36. (c) Use the probability of the complement from part (b), P (Ac) = 1/36, and the equation for the complement: P (less than 12) = 1 − P (12) = 1− 1/36 = 35/36.

16(a) First find P (6) = 5/36, then use the complement: P (not 6) = 1− P (6) = 31/36. (b) First find the complement, which requires much less effort: P (2 or 3) = 1/36 + 2/36 = 1/12. Then calculate

P (B) = 1− P (Bc) = 1− 1/12 = 11/12. (c) As before, finding the complement is the clever way to determine P (D). First find P (Dc) = P (11 or 12) =

2/36 + 1/36 = 1/12. Then calculate P (D) = 1− P (Dc) = 11/12.