We can also calculate the probability, , of the needle crossing a line as the product of and , where is the probability that the center of the needle falls close enough to a line to possibly cross it and is the probability that the needle actually crosses the line, given that its center is within reach.
Let represent the length of the needle and represent the width of each piece of wood (that is, the distance between two lines).
The needle can possibly cross a line if its center is within of either side of the line. So, adding to account for the needle falling on either side of the line, then dividing by the total distance between this line and the next, , we get
Now, we assume the center is within reach of crossing a line, meaning it lies or less from a line.
Recall that the needle will cross a line for a given when , or . The probability of this happening is thus , since we assume ranges uniformly between 0 and , independently of . Taking the average overall possible values of between 0 and , we find that:
Putting this all together, we obtain .