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.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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