Question

In the figure, a sector of the circle with central angle 120° is given. If a dot is put in the circle without looking, what is the probability that the dot is in the shaded region ? 

 

Hint:

In geometric probability problems involving sectors of a circle, you don't need to calculate the actual areas using π r². The ratio of the areas is the same as the ratio of their central angles, which simplifies the calculation significantly.

Answer 1

The question asks for the geometric probability of a randomly placed dot falling within the shaded region of a circle. The shaded region is the major sector, and the unshaded region is the minor sector with a central angle of 120°.

Geometric probability is calculated as the ratio of the favorable area to the total area.
Probability = Area of the favorable regionTotal Area Since the area of a sector is directly proportional to its central angle, we can also use the ratio of the angles.
Probability = Central angle of the favorable regionTotal angle in a circle The total angle in a circle is 360°.
The central angle of the unshaded minor sector is given as 120°.
The shaded region is the major sector. Its central angle is the total angle minus the angle of the minor sector.
Angle of shaded region = Total angle - Angle of unshaded region
Angle of shaded region = 360^ - 120^ = 240^ This is the favorable angle.
Now, we can calculate the probability:
P(dot in shaded region) = Angle of shaded regionTotal angle in a circle P(dot in shaded region) = (240^)/(360^) To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 120.
P(dot in shaded region) = (240 ÷ 120)/(360 ÷ 120) = (2)/(3) The probability that the dot is in the shaded region is (2)/(3).