Question

A table is ruled with equidistant parallel lines a distance 20 inches apart. A needle of length 4 inches, is randomly thrown on the table. What is the probability that the needle will intersect one of the lines (the other possibility being that the needle will be completely contained in the strip between two lines)?

Answer 1

Let y denote the distance from the centre of the needle to the nearest parallel line. & denotes the angle that the needle maken with the parallel line. Here, I ranges from oto 10 and oranges from oto Tl. 20 inches Ginches OY Since the needle is randomly Thrown on the table, all possible valuer of Yand o are taken as equally likely. The joint probability density function of y and o is given by osyç 10 This is the pdf of a fly,0)=k; oso si uniform distribution and kis a constant. By using trigromelic equations from the diagram, we get y = - (4 sin o) , the needle will intersect one of the parallel lines if yis less than 1 lusind) e 2 sind

We know that probabilla (o)= No. of favourable cases Total No. of cases. Here, favorable favorab care in when there is chance of the needle intersecting the parallel line, this will happen when y ranges from oto 2 sine and ocosa So, The probability can be written as, T 2Bino S S fly, o) dy do T fly,o) dy do stono sino do 10 T toma [-so) ((---3]