Question

A horizontal plane is ruled with parallel lines 5 inches apart. A five inch needle is tossed randomly onto the plane. The probability that the needle will touch a line is given below,
P = 2/π ∫π/2 0 sinθ dθ
where θ is the acute angle between the needle and any one of the parallel lines.
1. Find this probability. (Round your answer to one decimal place.)

Answer 1

Answer: 0.6

Step-by-step explanation:

To get the probability P, we need to evaluate the definite integral given.

P = 2/π ∫π/2 0 sinθ dθ

Integration of sinθ is -cosθ

P = 2/Π{-cosθ}... (1)

Substituting the upper limit of the integral which is Π/2 into equation 1;

P1 = 2/Π{-cosΠ/2}

P1 = 2/Π{0} since cosΠ/2 is "zero"

Similarly substituting the value at the lower limit which is "zero" into equation (1)

P2 = 2/Π{-cos0}

P2 = 2/Π{-1} since cos0 is 1

P2 = -2/Π

The probability P will be equal to P2-P1 {upper limit value - lower limit value}

P = 0-(-2/Π)

P = 2/Π

P = 0.6 ( to 1d.p)