Geometric Probability

– Solve problems involving geometric probability– Solve problems involving sectors and segments of circles

To win at darts, you must throw the darts into the part of the dartboard that earns the most points.

Probability that involves a geometric area such as length or area is called geometric probability.

GEOMETRIC PROBABILITY

Key Concept Probability and Area

A

B

If a point in region A is chosen at random, then the probability P(B) that the point is in region B, which is in the interior of region A, is

P(B) =area of region Barea of region A

Example 1 Probability with Area

A square gameboard has blue and white stripes of equal width as shown. What is the chance that a dart thrown at the board will land on a white stripe?

Example 1 Probability with Area

A square gameboard has blue and white stripes of equal width as shown. What is the chance that a dart thrown at the board will land on a white stripe?

Extend the sides of each stripe. This separates the square into 36 small unit squares.The white stripes have an area of 15 square units.The total area is 36 square units.The probability of tossing a dart into the white stripes is or15

365

12

SECTORS AND SEGMENTS OF CIRCLES

A sector of a circle is a region of a circle bounded by a central angle and its intercepted arc.

Central Angle

Arc

Sector

Key Concept Area of a SectorIf a sector of a circle has an area of A square units, a central angle measuring N°, and a radius of r units,

then A = r2

N°r

N360

Example 2 Probability with Sectors

° °

°

°°

°

a) Find the area of the blue sector.

12

b) Find the probability that a point chosen at random lies in the blue sector.

chord

arc

segment

The region of a circle bounded by an arc and a chord is called a segment of a circle. To find the area of a segment, subtract the area of a triangle formed by the radii and the chord from the area of the sector containing the segment.

Example 3 Probability with Segments

14

A regular hexagon is inscribed into a circle with a diameter of 14.

a)Find the area of the red segment.

b) Find the probability that a point chosen at random lies in the red segment.