Points in a circle

Probability Level 3

$N^2$ points are chosen uniformly and at random inside a circle of radius $N$ , for some positive integer $N$ .

For large $N$ , i.e. $N \to \infty$ , what is the probability that exactly one point will be within one unit of the center of the circle?

Provide your answer to three decimal places.

The answer is 0.368.

1 solution

Geoff Pilling
Aug 6, 2017

A circle of radius $N$ has an area of $\pi N^2$ , and in order to be within one unit from the circle, a point must be inside a circle at the origin of area $\pi$ .

Therefore, each point has the probability $\frac{1}{N^2}$ of being inside the circle at the origin, and $\frac{N^2-1}{N^2}$ of being outside the circle.

So, since there are $N^2$ points to consider, the probability is given by:

$P = N^2 \frac{1}{N^2}(\frac{N^2-1}{N^2})^{N^2-1}$

$P = (\frac{N^2-1}{N^2})^{N^2-1}$

For very large $N$ ,

$P = \lim_{N \to \infty} {(\frac{N^2-1}{N^2})}^{N^2-1} = \frac{1}{e} = 0.368$

How did you get $P = N^2 \frac{1}{N^2}(\frac{N^2-1}{N^2})^{N^2-1}$ ?

Ron Lauterbach - 3 years, 8 months ago

Would that be the answer if you were to ask the probablity of exactly 2 points to be within one unit of the center of the circle? $P = \lim_{N \to \infty} {{N^2 \choose 2}(\frac{1}{N^2})^2(\frac{N^2-1}{N^2})}^{N^2-2}$

Tolga Gürol - 2 years, 5 months ago

Points in a circle

The answer is 0.368.

1 solution

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