10. Needles and Bins A needle of length 2a is dropped uniformly at random on a surface ruled with parallel vertical lines separated by distance 2a and parallel horizontal lines separated by a distance of 2a. In class we derived the probability p that the needle crosses a vertical line and the probability P2 that the needle crosses a horizontal line. (a) Determine the probability p3 that the needle crosses a vertical line and a horizontal line. (Hint: Drawing a picture may help. Your answer should turn out to be 1/.) (b) Determine the probability p4 that the needle crosses either a vertical line or a horizontal line, or both. Express your answer numerically and as a function of P1, P2, P3. (c) Determine the probability p5 that the needle crosses a vertical line given that it crosses a horizontal line. Express your answer numerically and as a function of P1, P2, P3. Is P5 larger or smaller than p? a 0 -a -a 0 a The following identities may be useful to you: sin( T T -0) = cos(0) cos( -0) = sin(0) 2 sin(20) 2 sin(0) cos(0) cos(20) = cos (0)- sin (0)