Related Topics: More Probability Lessons

In these lessons, we will learn how to solve probability problems that may involve geometry and the area of 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 circle =

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

**Geometric Probability with Area**

Example 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 the dart lands inside the circle? Give the exact probability and the probability as a percent rounded to the nearest tenth.

Example 2:

A point is chosen at random on the given figure. What is the probability that the point is in the yellow region?

Example 3:

A square is inscribed in a circle. What is the probability that a point chosen inside the circle will be inside the square?

Example 4:

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

**Using Area to find Probability**

Example:

A circle is inscribed in a square. Point Q in the square is chosen at random. What is the probability that Q lies in the shaded region?

**Geometric Probability and Areas of Sectors**

Geometric Probability is probability that involves geometric measures (typically length or area)

Example:

Find the area of the indicated sector. Then find the probability of spinning the color indicated if the diameter is 12 cm.

**Study Guide Area Probability Problem 1**

Example:

Find the probability that a point randomly selected from the figure will land in the shaded area.

**Study Guide Area Probability Problem 2**

Example:

Find the probability that a point randomly selected from the figure will land in the shaded area.

**Geometric Probability**

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