# Needle on parallel lines, problem

• mkkrnfoo85
In summary, the problem given is to compute the probability that a randomly tossed needle of length 1 inch will touch one of the parallel East-to-West lines that are spaced 2 inches apart on the ground. There are various ways to approach this problem, such as considering the circles generated by spinning the needle or defining the random variable and probability distribution. This is a classic problem that has been used to estimate mathematical constants.

#### mkkrnfoo85

Hi, I was given a problem by the professor, and I feel like I do not know where to begin. Well, here is the problem:
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Endless, parallel East-to-West lines are spaced 2 inches apart on the ground, and a needle of length 1 inch is randomly tossed on the ground.
Compute the probability that the needle touches a line.
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I have tried attempting it, but I really don't have anywhere to start. Would anyone perhaps nudge me in the right direction? Thanks in advance.

-Mark

This is a classical problem with a very interesting solution. There's a couple ways of doing it. Think about circles that spinning the needle generates...

One thing you will need to decide is: what is your random variable and what probability distribution will you use? It is probably simplest to assume a uniform distribution but what variable? The position of the center of the needle or one end point? The angle the needle makes with the horizontal? A combination of those?

ah thanks guys.

oh yeah nice problem, a classic one too... I won't spoil the fun but just do some geometry. In fact... I remember that there is like this one guy that does this experiment over and over and over... and over again to estimate a certain mathematical constant.

tim_lou said:
oh yeah nice problem, a classic one too... I won't spoil the fun but just do some geometry. In fact... I remember that there is like this one guy that does this experiment over and over and over... and over again to estimate a certain mathematical constant.

And then fudged his results before publishing them, of course.

;0

## 1) What is the needle on parallel lines problem?

The needle on parallel lines problem, also known as Buffon’s needle problem, is a mathematical problem that involves randomly dropping a needle onto a set of parallel lines and calculating the probability that the needle will cross one of the lines.

## 2) What is the significance of this problem?

This problem has been used to demonstrate the concept of randomization and to calculate the value of pi. It also has practical applications in fields such as statistics and physics.

## 3) What assumptions are made in this problem?

This problem assumes that the needle is much shorter than the distance between the parallel lines, and that the needle is dropped randomly and uniformly.

## 4) How is the probability of the needle crossing a line calculated?

The probability is calculated by dividing the length of the needle by the distance between the lines, and then multiplying it by 2/pi. This formula is derived from the mathematical concept of geometric probability.

## 5) What are some variations of this problem?

Some variations of this problem include using a different shape instead of a needle, such as a square or triangle, and dropping the object onto a grid of parallel lines instead of just two lines. These variations can provide insights into different mathematical concepts and applications.