Probability involving a needle

In summary, the conversation discusses different approaches to solving the Buffon's needle problem, with the main idea being to calculate the probability that the needle hits a line when randomly dropped on a grid of parallel lines. The conversation mentions using a double integral to find the probability, but also introduces a simpler approach using the angle between the needle and the lines. The solution on Wikipedia involves transforming the needle into a noodle and using linearity to calculate the probability. Ultimately, the probability is found to be 2/π, regardless of the shape or length of the needle.
  • #1
LCSphysicist
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TL;DR Summary
How you would approach this problem? We need to find the accesibles position of a dropping needle
1603288012980.png

My approach to this problem is a little laborious, it involves three coordinates, probably it is right, but tiring and extensive beyond what the question wanted.

Be the origin in the rectangle middle.

It would be like: imagine a rectangle with opposite sides L and R with length l, so to find the probability the needle cross a line, i would at first find the probability the needle center fall in a position between x,x + dx; y, y + dy, so find the allowed angle to it rotate without cross a line, find the probability it falls in this range (range/2pi), ant integrate wrt dx,dy, finding Pf. But, as what we want is exactly opposite of it, we would just do P = 1 - Pf

In another words:

$$P = 1- \int\int P((x,y))P(\theta)dxdy$$

where we can express $$\theta = \theta(y)$$ (what matters to the angle is y itself, with it we can know the distance between the line and the needle center as l-y

In the end the x value will cancel $P(x,y) = \frac{dxdy}{lX}$, when we integrate dx, it will just cancel with X) and we will finish with just y.

But i believe maybe there is an easier way to do it... What would you do?
 
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  • #4
Can you just do something like the angle ##\theta## between the needle and the lines is uniformly distributed between 0 and ##\pi##. ##\theta=0## means the needle is parallel. Then the length covered by the needle in the direction perpendicular to the lines is ##l \sin(\theta)##, and the probability it hits a line is ##l \sin(\theta) / l##.

Then the probability it hits a line is
$$\frac{1}{\pi} \int_{0}^{\pi} \sin(\theta) d\theta $$
Which yields ##\frac{2}{\pi}##.

This is the same as what you wrote down, but picking a slightly better perspective to avoid a double integral.

Edit to add: wikipedia has an amazing solution for this problem:
https://en.m.wikipedia.org/wiki/Buffon's_noodle

Basically the idea is:. If you curve the needle and make it a noodle (of any rigid shape), the expected number of line crossings is independent of the shape, since you can think of that shape as being broken up into small linear chunks and then observing that expectancy adds linearly. It also must be linear in the length of the noodle. If you take a circle of diameter l, it will always intersects exactly twice, so the rate of intersections is 2 per length ##\pi l##. Hence if the noodle is a needle of length ##l##, the expected value of the number of intersections is ##l \frac{2}{\pi l} = \frac{2}{\pi}##. Since it only ever intersects 0 or 1 times, that must be the probability that it intersects.
 
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Related to Probability involving a needle

What is the concept of probability involving a needle?

The concept of probability involving a needle is based on the likelihood or chance of an event occurring. In this case, the needle represents a random event and the probability is the numerical measure of the likelihood that the needle will land in a specific position or area.

How is probability calculated for a needle?

The probability of a needle landing in a specific position or area can be calculated by dividing the total number of possible outcomes by the total number of desired outcomes. This can be represented as a fraction, decimal, or percentage.

What factors affect the probability of a needle?

The probability of a needle landing in a specific position or area can be affected by various factors such as the length of the needle, the width of the spaces it can land in, and the orientation of the needle when it is dropped. These factors can change the total number of possible outcomes and therefore, impact the probability.

How can probability involving a needle be applied in real life?

Probability involving a needle can be applied in various fields such as statistics, gambling, and physics. It can be used to predict the likelihood of an event occurring and make informed decisions based on those predictions.

What are some common misconceptions about probability involving a needle?

One common misconception is that the probability of a needle landing in a specific position is always 50/50, but this is not always the case. The probability can vary depending on the factors mentioned earlier. Another misconception is that probability can be used to predict the outcome of a single event, but it is actually a measure of likelihood over a large number of trials.

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