2 Objectives: Solve problems involving geometric probability. Solve problems involving sectors and segments of circles.
3 [ Probability is expressed as a fraction, decimal, or percent.] Geometric Probability- Probability that involves a geometric measure such as length or area.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 isP(B)= area of region Barea of region AAB
4 Example 1:What is the chance that a dart will land in the yellow area of the grid?Find the area of the whole grid.To find the area of the yellow area, count the square units.Yellow area: 28Grid area: 49P(yellow)= 2849Which would be: 4/7 or ≈.6
5 Example 1 continued: What is the chance that a dart will land in the yellow area of the grid? Find the area of the whole grid. (7 times 7) To find the area of the yellow area, count the square units. Yellow area: 28 Grid area: 49 P(yellow)= Which would be: 4/7 or ≈.6
6 Find the probability that a point chosen at random Your Turn:Find the probability that a point chosen at randomwill land in the green area.Region A:Region B:
7 How to solve: First, we have to find the area of region A. Then, we find the area of region B.P(B)= 981So P(B)= 19So therefore, the probability that a point will land in region B is 1/9 or ≈ .11
8 - A sector of a circle is a region of a circle bounded by a central angle and its intercepted arc.
9 Area of a SectorIf a sector of a circle has an area of A square units, a central angle measuring N degrees, and a radius of r units, thenSectorN
10 Probability with Sectors We know how to calculate the area of asector but we still have to find the probabilitythat a point chosen at random will land inthat sector. The way you do this is by usingthe following formula:P = area of sectorarea of circle
11 Example 2: Find the area of sector XYZ and then find the probability that a point chosen at random will land in sector XYZ.XYZ
12 How To: Put the central angle over 360 and multiply it to pi and r². A= 26.2cm²To find the probability:Area of a circle is πr².So, π(5²)= π(25)*calculator*Area of circle is ≈78.5cm²P = 26.278.5Probability that a random point is in sectorXYZ is about .33 or 33%
13 Find the measure of X, the area of the sector ABC, and find the probability that a point chosen at random will land in sector ABC.X degrees= 57°Area of sector: 57 (3.14)(8²)360A≈ 31.8m²Area of circle: (3.14)(8²)A≈P = 31.8200.9616mABX degreesX = 57A =31.8m²P ≈ .16123°C
14 The region of a circle bounded by an arc and a chord is called a segment of a circle.
15 Area/Probability of a Segment In any regular polygon inscribed in a circle, to find thearea of a segment, you first have to:Find the area of a sector.Find the area of a triangle.Then, you can find the area of the segment.Area of Segment= area of sector- area of triangleTo find the probability that a point chosen at random lies in the segment is:P = area of segmentarea of circle
16 Example 3:Find the area of one of the blue segments in the circle and the probability that a point chosen at random will land in the segment.Area of sector: 60 (π)(3²)360≈ 4.7 cm²Area of triangle: (1/2)(bh)(1/2)(3)(h)The are equilateral triangles,so the apothem would make a30°-60°-90° triangle.Therefore, the height would be1.5√3.A = (1/2)(3)(1.5√3)= ≈ 3.9cm²A of segment= 4.7 – 3.9A = 0.8cm²Area of circle: (3.14)(3²)≈ 28.3P = 0.828.3 ≈.03 or 3%6cm30°1.5√360°1.5cm1.5cm
17 Each central angle is 120°Find the area of the shaded region. Then, find the probability that a point chosen at random will land in the green area.Area of Sector: 120 πr²360A = 120/360 (3.14)(15²)= 235.5in²Apothem= 7.5Base of triangle= 7.5√3≈13inArea of triangle:½(26)(7.5)=97.5in²30 in.Area of circle:15²(3.14)= 706.5in²15in.60°7.5 in.Area of Segment:235.5 – 97.5 = 138in²30°P = 138706.513 in.≈.2 or 20%