Related Topics: More examples of Probability

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

**Probability of shaded region**

^{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 =

b) in the green sector.

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

**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** =

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 = 2x and QR = 2y. Then, AQ = x and QB = y.

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

Area AQB = xy

**Probability of striking triangle AQB** = xy ÷ 4xy =

b) All the shaded triangles are equal.

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

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

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

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

Area of triangle BRC = xy

Probability of striking triangle BRC =

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

**Geometric Probability using Area**

Example 1: A circle with radius 2 lies within 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 tenths.

Example 2: A point is chosen at random on this figure. What is the probability that the point is in the yellow region?

Example 3: A square is inscribed inside a circle. What is the probability that a point chosen at random inside the circle will be inside the square?

Example 4: A circle is inscribed in an equilateral triangle. What is the probability that a point chosen at random in the triangle will be inside the circle?

Problem 1: Find the probability that a point chosen at random inside the circle will be inside the shaded region.

Problem 2: Find the probability that a point chosen at random inside the square will be inside the shaded region.**Area Probability Problem: Rectangle within a rectangle**

Example: Find the probability that a point randomly selected from a figure would land in the shaded area. Area Probability Problem 2 Area Probability Problem 3

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

** 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: **

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

Area of shaded region = 615.83 – 153.99 = 461.84 cm

Probability of hitting the shaded region =

** Example: **

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

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

a) in the red sectorb) in the green sector.

c) in any sector

** Solution:**

b) Area of green sector = × area of circle

c) in any 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.

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

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

Area AQB = xy

b) All the shaded triangles are equal.

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

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

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

Area of triangle BRC = xy

Probability of striking triangle BRC =

Example 1: A circle with radius 2 lies within 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 tenths.

Example 2: A point is chosen at random on this figure. What is the probability that the point is in the yellow region?

Example 3: A square is inscribed inside a circle. What is the probability that a point chosen at random inside the circle will be inside the square?

Example 4: A circle is inscribed in an equilateral triangle. What is the probability that a point chosen at random in the triangle will be inside the circle?

Problem 1: Find the probability that a point chosen at random inside the circle will be inside the shaded region.

Problem 2: Find the probability that a point chosen at random inside the square will be inside the shaded region.

Example: Find the probability that a point randomly selected from a figure would land in the shaded area. Area Probability Problem 2 Area Probability Problem 3

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