Probability
Geometrical Probability
Grade 12
Question:
<p>A point is selected at random from the interior of a circle. The probability that the point is closer to the centre than boundary of the circle is</p>
<p>(a) \(\frac{3}{4}\)</p>
<p>(b) \(\frac{1}{2}\)</p>
<p>(c) \(\frac{1}{4}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the geometric probability formula where the probability equals the ratio of areas. A point closer to the center than the boundary lies within a circle of radius r/2.
<p><strong>Solution:</strong> In a circle, all the points which lie in the area of half the radius will be nearer to origin than to the boundary.</p><p>$n(S) = $ The area of circle of radius $r$</p><p>$n(E) = $ The area of the circle of radius $\frac{r}{2}$</p><p>Required probability $= \frac{n(E)}{n(S)} = \frac{\pi(r/2)^2}{\pi r^2} = \frac{1}{4}$</p>
Correct Answer: c