Geometry

Geometric Probability

April 20, 2023

Goals

Know what probability is. Use areas of geometric figures to

determine probabilities.

April 20, 2023

Probability

A number from 0 to 1 that represents the chance that an event will occur.

P(E) means “the probability of event E occuring”.

P(E) = 0 means it’s impossible. P(E) = 1 means it’s certain. P(E) may be given as a fraction,

decimal, or percent.

April 20, 2023

Probability

Number of Successful OutcomesP(E)=

Total number of Outcomes

Example

A ball is drawn at random from the box. What is the probability it is red?

P(red) = ??29

April 20, 2023

Probability

Number of Successful OutcomesP(E)=

Total number of Outcomes

A ball is drawn at random from the box. What is the probability it is green or black?

P(green or black) = ??39

April 20, 2023

Probability

Number of Successful OutcomesP(E)=

Total number of Outcomes

A ball is drawn at random from the box. What is the probability it is green or black?

P(green or black) = 13

April 20, 2023

Geometric Probability

Based on lengths of segments and areas of figures.

Random:Without plan or order. There is no bias.

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Probability and Length

Let AB be a segment that contains the segment CD. If a point K on AB is chosen at random, then the probability that it is on CD is

Length of CDP(K is on CD)

Length of AB

April 20, 2023

Example 1

Find the probability that a point chosen at random on RS is on JK.

JK = 3

RS = 9

Probability = 1/3

1 2 3 4 5 6 7 8 9 10 11 12

R SJ K

April 20, 2023

Your Turn

Find the probability that a point chosen at random on AZ is on the indicated segment.

15

25

12

AB

AC

BD

1 2 3 4 5 6 7 8 9 10 11 12

A ZB C D E

910

110

45

AE

EZ

BZ

April 20, 2023

Probability and Area

Let J be a region that contains region M. If a point K in J is chosen at random, then the probability that it is in region M is

Area of MP(K is in M)=

Area of J

MJ

K

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Example 2

Find the probability that a randomly chosen point in the figure lies in the shaded region.

8

8

April 20, 2023

Example 2 Solution

8

8

Area of Square = 82 = 64

Area of Triangle

A=(8)(8)/2 = 32

Area of shaded region

64 – 32 = 32

Probability:

32/64 = 1/2

8

April 20, 2023

Example 3Find the probability that a randomly chosen point in the figure lies in the shaded region.

5

April 20, 2023

Example 3 Solution

5510

Area of larger circle

A = (102) = 100

Area of one smaller circle

A = (52) = 25

Area of two smaller circles

A = 50

Shaded Area

A = 100 - 50 = 50

Probability50 1

100 2

April 20, 2023

Your TurnA regular hexagon is inscribed in a circle. Find the probability that a randomly chosen point in the circle lies in the shaded region.

6

April 20, 2023

Solution

3 3

121

3 3 362

54 3 93.53

A ap

6 ?6

?3?3 3

Find the area of the hexagon:

April 20, 2023

Solution

3 36 6

33 3

Find the area of the circle:

A = r2

A=36 113.1

Shaded Area

Circle Area – Hexagon Area

113.1 – 93.63 =19.57

113.1

19.57

93.53

April 20, 2023

Solution

3 36 6

33 3

Probability:

Shaded Area ÷ Total Area

19.57/113.1 = 0.173

17.3%113.1

19.57

April 20, 2023

Example 4

If 20 darts are randomly thrown at the target, how many would be expected to hit the red zone?

10

April 20, 2023

Example 4 Solution

10

Radius of small circles:

5

Area of one small circle:

25

Area of 5 small circles:

125

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Example 4 Solution continued

10

Radius of large circle:

15

Area of large circle:

(152) = 225

Red Area:

(Large circle – 5 circles)

225 125 = 100

10

5

April 20, 2023

Example 4 Solution continued

10

Red Area:100

Total Area: 225

Probability:

100 40.444...

225 9

This is the probability for each dart.

April 20, 2023

Example 4 Solution continued

10

Probability:

100 40.444...

225 9

For 20 darts, 44.44% would likely hit the red area.

20 44.44% 8.89, or about 9 darts.

April 20, 2023

Your Turn

500 points are randomly selected in the figure. How many would likely be in the green area?

5 3

April 20, 2023

Solution500 points are randomly selected in the figure. How many would likely be in the green area?

5 3

10

Area of Hexagon:

A = ½ ap

A = ½ (53)(60)

A = 259.81

Area of Circle:

A = r2

A = (53)2

A= 235.62

60

30

5

10

April 20, 2023

Solution500 points are randomly selected in the figure. How many would likely be in the green area?

5 3

Area of Hexagon:

A = 259.81

Area of Circle:

A= 235.62

Green Area:

259.81 – 235.62

24.19

April 20, 2023

Solution500 points are randomly selected in the figure. How many would likely be in the green area?

5 3

Area of Hexagon:

A = 259.81

Green Area:

24.19

Probability:

24.19/259.81 =

0.093 or 9.3%

April 20, 2023

Solution500 points are randomly selected in the figure. How many would likely be in the green area?

5 3

Probability:

0.093 or 9.3%

For 500 points:

500 .093 = 46.5

47 points should be in the green

area.

April 20, 2023

Summary

Geometric probabilities are a ratio of the length of two segments or a ratio of two areas.

Probabilities must be between 0 and 1 and can be given as a fraction, percent, or decimal.

Remember the ratio compares the successful area with the total area.

April 20, 2023

Practice Problems