geometry geometric probability. october 25, 2015 goals know what probability is. use areas of...

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

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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

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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

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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

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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

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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

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Solution

3 36 6

33 3

Probability:

Shaded Area ÷ Total Area

19.57/113.1 = 0.173

17.3%113.1

19.57

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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