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Geometry

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Geometry 11.6 Geometric Probability Goals Know what probability is. Use areas of geometric figures to determine probabilities. Probability A number from 0 to 1 that ... – PowerPoint PPT presentation

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Title: Geometry


1
Geometry
  • 11.6 Geometric Probability

2
Goals
  • Know what probability is.
  • Use areas of geometric figures to determine
    probabilities.

3
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 its impossible.
  • P(E) 1 means its certain.
  • P(E) may be given as a fraction, decimal, or
    percent.

4
Probability
Example A ball is drawn at random from the box.
What is the probability it is red?
?
2
P(red)
?
9
5
Probability
A ball is drawn at random from the box. What is
the probability it is green or black?
?
3
P(green or black)
?
9
6
Probability
A ball is drawn at random from the box. What is
the probability it is green or black?
1
P(green or black)
3
7
Geometric Probability
  • Based on lengths of segments and areas of
    figures.
  • Random
  • Without plan or order. There is no bias.

8
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

9
Example 1
  • Find the probability that a point chosen at
    random on RS is on JK.

JK 3 RS 9 Probability 1/3
10
Your Turn
  • Find the probability that a point chosen at
    random on AZ is on the indicated segment.

11
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

J
M
K
12
Example 2
  • Find the probability that a randomly chosen point
    in the figure lies in the shaded region.

8
8
13
Example 2 Solution
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
8
8
14
Example 3
  • Find the probability that a randomly chosen point
    in the figure lies in the shaded region.

5
15
Example 3 Solution
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?
5
5
10
Probability
16
Your Turn
  • A regular hexagon is inscribed in a circle. Find
    the probability that a randomly chosen point in
    the circle lies in the shaded region.

6
17
Solution
Find the area of the hexagon
?
6
?
?
3
18
Solution
19.57
Find the area of the circle A ?r2 A36? ?
113.1 Shaded Area Circle Area Hexagon
Area 113.1 93.63 19.57
93.53
6
113.1
3
19
Solution
19.57
Probability Shaded Area Total Area 19.57/113.1
0.173 17.3
6
113.1
3
20
Example 4
  • If 20 darts are randomly thrown at the target,
    how many would be expected to hit the red zone?

10
21
Example 4 Solution
Radius of small circles 5 Area of one small
circle 25? Area of 5 small circles 125?
10
22
Example 4 Solution continued
Radius of large circle 15 Area of large
circle ?(152) 225? Red Area (Large circle 5
circles) 225? ? 125? 100?
10
5
10
23
Example 4 Solution continued
Red Area100? Total Area 225? Probability
10
This is the probability for each dart.
24
Example 4 Solution continued
Probability
10
For 20 darts, 44.44 would likely hit the red
area. 20 ? 44.44 ? 8.89, or about 9 darts.
25
Your Turn
  • 500 points are randomly selected in the figure.
    How many would likely be in the green area?

26
Solution
  • 500 points are randomly selected in the figure.
    How many would likely be in the green area?

Area of Hexagon A ½ ap A ½ (5?3)(60) A
259.81 Area of Circle A ?r2 A ?(5?3)2 A
235.62
10
27
Solution
  • 500 points are randomly selected in the figure.
    How many would likely be in the green area?

Area of Hexagon A 259.81 Area of Circle A
235.62 Green Area 259.81 235.62 24.19
28
Solution
  • 500 points are randomly selected in the figure.
    How many would likely be in the green area?

Area of Hexagon A 259.81 Green
Area 24.19 Probability 24.19/259.81 0.093 or
9.3
29
Solution
  • 500 points are randomly selected in the figure.
    How many would likely be in the green area?

Probability 0.093 or 9.3 For 500 points 500 ?
.093 46.5 47 points should be in the green
area.
30
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.
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