There is a point in a circle. what is the probability that this point is closer to its circumference

Option 1 : $\frac{1}{4}$

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Calculation

Consider two concentric circles as shown in the following figure:

Radius of the inner circle $= {\rm{}}\frac{{\rm{r}}}{2}$

Radius of the outer circle = r

Points lying inside the inner circle will definitely be closer to the centre than the circumference of the outer circle.

Possible region where these points lie = area of the inner circle = $\frac{{{\rm{\pi }}{{\rm{r}}^2}}}{4}$

Required probability $= \frac{{{\rm{area\;of\;the\;inner\;circle}}}}{{{\rm{total\;area}}}}$

$= \frac{{{\rm{\pi }}{{\rm{r}}^2}}}{{4\; \times \;{\rm{\pi }}{{\rm{r}}^2}}}$

$= \frac{1}{4}$

Correct option is (1).

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As the comments point out, the question doesn't make much sense as written.

If the book means "circumference", then you're right, just compare the areas of the two circles. Note that your expression can be simplified to just $\frac14$. This is the same question: A point in a circle is selected at random. Calculate probability that point is closer to centre than circumference

If the book really means "a given radius", then the region of points closer to the center than to the rest of the radius would be the half-circle opposite from the given radius, and the answer would be $\frac12$.

Since the back of the book says $\frac14$, it seems that it meant to say "circumference".

You can put this solution on YOUR website!

There is a point in a circle. What is the probability that this point is closer to its circumference than to the centre? ------- Let the radius of the circle be "r". Total area of the circle = pi*r^2 -------- Radius of the circle with the same center and radius = r/2. Area of this smaller circle is pi(r/2)^2 = (1/4)pi*r^2 ----- Area between the smaller and the larger circle is (3/4)pi*r^2 ---- P(closer to circumference then to center) = (3/4) ------------------- Cheers, Stan H.

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Answer

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Hint: Circumference of the circle or perimeter of the circle is the measurement of the boundary of the circle. Whereas the area of the circle defines the region occupied by it. Area of any circle is the region enclosed by the circle itself or the space covered by the circle. The formula to find the area of the circle is = \[\pi {r^2}\] and for point to be close to the circumference than to the centre the point must be at a distance greater than \[\dfrac{r}{2}\] from the centre of circle.

Complete step by step solution:

Total possible outcomes = Area of circle = \[\pi {r^2}\]And let the radius of the circle be 'r'. Observe the figure, we have to find the probability of point, P in the ring which will be closer to circumference. For point to be close to the circumference than to the centre the point must be at a distance greater than \[\dfrac{r}{2}\] from the centre of circle where r is the radius, in which in the figure \[r = \dfrac{r}{2}\].

Area of ring = Area of outer circle - Area of inner circleArea of ring = \[\pi {r^2} - \pi {\left( {\dfrac{r}{2}} \right)^2}\]Area of ring= \[\pi {r^2} - \dfrac{{\pi {r^2}}}{4}\]Therefore, the outcome is \[\dfrac{{3\pi {r^2}}}{4}\].Area of region of a point such that distance is greater than \[\dfrac{r}{2}\] from the centre of circle = \[\dfrac{{3\pi {r^2}}}{4}\]Hence, the Probability that this point is close to the circumference than to the centre is\[\dfrac{{\dfrac{{3\pi {r^2}}}{4}}}{{\pi {r^2}}}\] = \[\dfrac{3}{4}\]

Therefore, option A is the right answer.

Note: The key point to find the probability in which the point is close to the circumference than to the centre, we can consider the area of the region of the point with respect to area of circle and the distance surrounding a circle is known as the circumference of the circle. The diameter is the distance across a circle through the centre, and it touches the two points of the circle perimeter.