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 2*x* be the length of the square.

Area of square = 2*x* × 2*x* = 4*x*^{2}

* 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 circl**e =

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

=

Example: Find the probability of a dart landing in the light purple region.

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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