11.5Geometric Probability

What is Geometric Probability?What is a sector of a circle?How to find the area of a sector of a circle?How to find the area of a segment of a circle?

Objectives

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

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 B Area of region A

What is Geometric Probability?

A B

Probability With Area

22

46

18

8

P(A)= Area of region A

AB

Area of region B

P(A)= 18 ∙ 8 46∙ 22

P(A)= .14~Simplify

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

What is a sector of a circle?

Central Angle

Arc

Sector

If 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 = (N/360)·πr²

Formula to find area of a Sector of a Circle

r

N°

Amount of money in sales each year

2008200920102011

How to find the area of a sector?

Interior Angle= 50

Interior Angle= 65

Interior Angle= 62 Interior Angle= 77

Find the area of 2010’s sales.

Area of any section= interior angle of section 360

radius2

A= 62 9

18

360~Radius= 9 Interior angle of 2010’s sales= 62

2

A= 43.8 units

2~ Simplify

Probability With Sectors

360

2radius

360A= 50 8

2~ Radius=8 Interior angle of section E= 50

Area= 8.9 ~ Simplify

Probability of any section= Area of section Area of

circleProbability of section E=8.9 82 ~ Area of section E= 8.9 Area of circle= 8

2

Probability of section E= 0.14 or 14%

Area of any section= interior angle of section

Interior Angle of Section A = 80Interior Angle of Section B = 57Interior Angle of Section C = 57Interior Angle of Section D = 46Interior Angle of Section E = 50Interior Angle of Section F = 70

A

BCD

E

F

Find the probability that a point chosen randomly will land in section E.

The region of a circle bounded by an arc and a chord is called a segment of a circle.

What is a segment of a circle?

Chord

Arc

Segment

Probability With Segments

A

A regular Hexagon is inscribed in a circle with a diameter of 14.What is the Probability that a point chosen randomly will land in segment A?

14

Area of the Sector:A= (N/360)·πr² =(60/360)·π(7²) =49π/6 =25.66

Area Of the triangle:Since the regular hexagon is inscribe in a circle the triangle is Equilateral, with each side 7 units long. Use Properties of 30°-60°-90° triangles to find the Apothem. The Apothem is 6.06.

7

Probability With Segments

14

A

7

Find Area of Triangle: A=1/2bh =1/2(7)(6.06) =21.22

Area of Segment = area of sector – area of triangle = 25.66 – 21.22 = 4.44 Find the probability: P(A) = Area of segment/Area of Circle = 4.44/153.94 = 0.03

The probability that a point, randomly selected, would land on segment A is about .03 or 3%

Pre-AP Geometry: Pgs. 625-627 #7-31

Geometry: Pg 625 #7 - 23

Assignment