Probability of Points Lying in a Disk

In summary, the probability of finding one or more points within a circle of radius r≤1 in a unit circle with M randomly placed points is (1-(1-r^2)^M), where the uniform density is 1/π and the probability of a uniform random point in S being in Cr is pi*r^2/A.
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marcusl
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Imagine points in a plane with a uniform density. To use a simple model, consider just a finite region such as the unit circle and randomly place M points into that unit circle. By uniform density, I mean that any infinitesimal area anywhere in the unit circle has the same probability of containing a point. How would I find the probability as function of M of finding one or more points within a circle of radius r≤1?

BTW, this is not a homework problem.
 
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In any subset, S, of finite area, A, the uniform density is 1/A. For a circle, Cr, of radius r <=1, the probability that a uniform random point in S is also in Cr is the ratio of the areas P = pi*r^2 / A. If the points are all independently placed, the odds that none are in Cr is (1-P)^M. The probability that at least one is in Cr is (1-(1-P)^M) For the set S = unit circle, P = r^2, so the answer is (1-(1-r^2)^M)
 
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Thanks. It seems so obvious once you explained it!
 

1. What is the definition of "Probability of Points Lying in a Disk"?

The probability of points lying in a disk refers to the likelihood of randomly chosen points falling within a given circular area, or disk. This probability can be calculated by dividing the number of points inside the disk by the total number of points in the sample space.

2. How is the probability of points lying in a disk calculated?

To calculate the probability of points lying in a disk, we first need to determine the area of the disk. This can be done by using the formula A = πr^2, where r is the radius of the disk. Then, we divide the number of points inside the disk by the total number of points in the sample space. This will give us a decimal value, which can be converted to a percentage to represent the probability.

3. What factors influence the probability of points lying in a disk?

The probability of points lying in a disk can be influenced by several factors, including the size and shape of the disk, the number of points in the sample space, and the distribution of the points within the sample space. Additionally, the presence of any obstacles or boundaries within the sample space can also affect the probability.

4. How does the number of points in the sample space affect the probability of points lying in a disk?

The number of points in the sample space directly affects the probability of points lying in a disk. As the number of points increases, the likelihood of randomly chosen points falling within the disk also increases. This is because the more points there are, the greater the chances of some of them falling within the circular area.

5. Can the probability of points lying in a disk be greater than 1?

No, the probability of points lying in a disk cannot be greater than 1. This is because the probability is a measure of the likelihood of an event occurring, and it cannot exceed 100%. If the calculated probability is greater than 1, it is likely that an error has been made in the calculation.

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