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Probability can also relate to the areas of geometric shapes. The following are some examples of probability problems that involve areas of geometric shapes.

Related Topics: More examples of Probability

* Example: *

A dart is thrown at random onto a board that has the shape of a circle as shown below. Calculate the probability that the dart will hit the shaded region. (Use π = 3.142)

* Solution: *

Total area of board = 3.142 × 14 2 = 615.83 cm^{2}

Area of non-shaded circle = 3.142 × 7 2 = 153.99 cm^{2}

Area of shaded region = 615.83 – 153.99 = 461.84 cm^{2} = 462 cm^{2} (rounded to whole number)

** Probability of hitting the shaded region: **

=

* Example: *

The figure shows a circle divided into sectors of different colours.

If a point is selected at random in the circle, calculate the probability that it lies:

a) in the red sector

b) in the green sector.

c) in any sector **except** the green sector.

* Solution:*

a) Area of red sector = × area of circle

**Probability that the point lies on red sector** =

b) Area of green sector = × area of circle

**Probability that the point lies on green sector** =

c) in any sector **except** the green sector.

**Probability that the point does not lie in the green sector** =

**Example:**

In the figure below, *PQRS* is a rectangle, and *A*, *B*, *C*, *D* are the midpoints of the respective sides as shown.

An arrow is shot at random onto the rectangle *PQRS*. Calculate the probability that the arrow strikes:

a) triangle *AQB*.

b) a shaded region.

c) either triangle *BRC* or the unshaded region.

* Solution: *

a) Let *PQ* = 2*x* and *QR* = 2*y*. Then, *AQ* = *x* and *QB *= *y*.

Area of rectangle *PQRS* = 2*x* × 2*y* = 4*xy*

Area *AQB* = *xy *

**Probability of striking triangle AQB** =

b) All the shaded triangles are equal.

Total area of shaded regions = 4 × *xy* = 2*xy*

**Probability of striking a shaded region** = 2*xy* ÷ 4*xy* =

c) Area of unshaded region = 4*xy* – 2*xy* = 2*xy *

Probability of striking unshaded region = 2*xy* ÷ 4*xy* =

Area of triangle *BRC* = *xy *

Probability of striking triangle *BRC*=

**Probability of striking triangle BRC or unshaded region** =

Geometric Probability with Area

Area Probability Problem 1

Area Probability Problem 2

Area Probability Problem 3