Suppose we throw a needle on a large sheet of paper, on which horizontal lines are drawn, which are at needle-length apart (see also Exercise XXXXXXXXXXChoose one of the horizontal lines as x-axis,...

Suppose we throw a needle on a large sheet of paper, on which horizontal lines are drawn, which are at needle-length apart (see also Exercise 21.16). Choose one of the horizontal lines as x-axis, and let (X, Y ) be the center of the needle. Furthermore, let Z be the distance of this center (X, Y ) to the nearest horizontal line under (X, Y ), and let H be the angle between the needle and the positive x-axis.

a. Assuming that the length of the needle is equal to 1, argue that Z has a U(0, 1) distribution. Also argue that H has a U(0, π) distribution and that Z and H are independent.

b. Show that the needle hits a horizontal line when

c. Show that the probability that the needle will hit one of the horizontal lines equals 2/π.