 # Geometric Probability

The following are more probability problems that involve geometrical shapes.

Example: ABCD is a square. M is the midpoint of BC and N is the midpoint of CD. A point is selected at random in the square. Calculate the probability that it lies in the triangle MCN.

Solution: Let 2x be the length of the square.

Area of square = 2x × 2x = 4x2  Example:

The figure shows a circle with centre O and radius 8 cm. Ð BOD = 72˚. The radius of the smaller circle is 4 cm. A point is selected at random inside the larger circle BCDE. Calculate the probability that the point lies
a) inside the sector BODC.
b) inside the smaller circle
c) neither in the sector BODC nor in the smaller circle.

Solution:

a) Area of sector BODC = × area of the large circle

Probability that the point lies in sector BODC = b) Area of smaller circle = × area of the large circle

Probability that the point lies in the smaller circle = c) Probability that the point does not lie in sector BODC or the smaller circle

= Geometric Probability
Example:
Find the probability of a dart landing in the light purple region. Geometric Probability Using Area
Examples:
(1) A circle with radius 2 lies inside a square with side length 6. A dart lands randomly inside the square. What is the probability that the dart lands inside the circle? Give the exact probability and the probability as a percent rounded to the nearest tenth.
(2) A point is chosen at random on this figure. What is the probability that the point is in the yellow region?
(3) A square is inscribed in a circle. What is the probability that a point chosen at random inside the circle will be inside the square?
(4) A circle is inscribed in an equilateral triangle. What is the probability that a point chosen at random inside the triangle will be inside the circle?
(5) Find the probability that a point chosen inside the circle will be inside the shaded region.

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