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(a)

From Consequence #1 in Problem 11. The probability that a random line passes through is , where is the perimeter of . Similarly, the probability that a random line passes through is . We are trying to find the probability that a line intersects only the larger set. Here, we are using conditional probability. We are only concerned with lines that intersect . Thus, our sample space is . The lines we seek are all the lines in the entire sample space, represented by the number 1, minus the lines that intersect . Finally, .

(b)

The probability of one random line was discovered in part (a). Each time an additional line is considered we multiply the probabilities together. So if n = 2, then . If we continue this pattern it becomes obvious that the probability that n random lines intersect and not is thus .

Problem 13