# Help solving a probability problem

I encountered the following problem. I am not very sure on how to solve it. Can someone help me solve the problem.

Problem: A dart is thrown onto a circular plate with unit radius. X is the random variable representing the distance of the point where the dart lands from the origin of the plate. Assume that the dart always lands on the plate and that the dart is equally likely to land anywhere on the plate.

Find (i) P(X < a) and (ii) P(a < X < b), where a < b <=1.

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You have a uniform distribution on the plate, so its density is $f_X(x)$ is $1/\pi$ on the plate and $0$ off the place. So if $A$ is a subset of the plate, $$P(A) = {|A|\over \pi},$$ where $|A|$ denotes the area of $A$.

Can you take it from here?

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My understanding of the problem is, since X is the distance of the dart from the origin, X is equally likely to take a value in the range [0,1]. So X has a uniform distribution with parameters (0,1). So its density function is f(x)=1 in [0,1] and 0 otherwise. So P(X<a) = a and P(a<X<b) =b-a. I could not see why area of the circular plate is considered here. Can you say whether my understanding is correct. –  Sujeeth Kumaravel Jan 1 '14 at 16:52
OP: Assume that the dart ... is equally likely to land anywhere on the plate. –  Did Jan 1 '14 at 22:07
Yes, the dart is equally likely to land anywhere on the plate. But, the random variable X just measures the distance from the origin. I do not understand how area comes into this picture. –  Sujeeth Kumaravel Jan 2 '14 at 20:21

(i) $P(X<a)=\frac{Area\; of\; circular\; plate \; with\; radius\; a}{Area\; of\; whole\; circular\; plate}=\frac{a^2\pi}{1^2\pi}=a^2$.

(ii) $P(a<X<b)=P(X<b)-P(X<a)$ gives a result.

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My understanding of the problem is, since X is the distance of the dart from the origin, X is equally likely to take a value in the range [0,1]. So X has a uniform distribution with parameters (0,1). So its density function is f(x)=1 in [0,1] and 0 otherwise. So P(X<a) = a and P(a<X<b) =b-a. I could not see why area of the circular plate is considered here. Can you say whether my understanding is correct. –  Sujeeth Kumaravel Jan 1 '14 at 16:54
OP: Assume that the dart ... is equally likely to land anywhere on the plate. –  Did Jan 1 '14 at 22:06