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A point is chosen at random inside a cir...

A point is chosen at random inside a circle. Find the probability that the point is closer to the centre of the circle than to its circumference.

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To find the probability that a point chosen at random inside a circle is closer to the center than to its circumference, we can follow these steps: ### Step 1: Understand the Problem We need to determine the area within a circle where points are closer to the center than to the circumference. ### Step 2: Define the Circle Let the radius of the larger circle be \( r \). ...
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There are some experiment in which the outcomes cannot be identified discretely. For example, an ellipse of eccentricity 2sqrt(2)//3 is inscribed in a circle and a point within the circle is chosen at random. Now, we want to find the probability that this point lies outside the ellipse. Then, the point must lie in the shaded region shown in Figure. Let the radius of the circle be a and length of minor axis of the ellipse be 2b. Given that 1 - (b^(2))/(a^(2)) = (8)/(9) or (b^(2))/(a^(2)) = (1)/(9) Then, the area of circle serves as sample space and area of the shaded region represents the area for favorable cases. Then, required probability is p= ("Area of shaded region")/("Area of circle") =(pia^(2) - piab)/(pia^(2)) = 1 - (b)/(a) = 1 - (1)/(3) = (2)/(3) Now, answer the following questions. A point is selected at random inside a circle. The probability that the point is closer to the center of the circle than to its circumference is

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