Title: Geometry
1Geometry
- 11.6 Geometric Probability
2Goals
- Know what probability is.
- Use areas of geometric figures to determine
probabilities.
3Probability
- 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.
4Probability
Example A ball is drawn at random from the box.
What is the probability it is red?
?
2
P(red)
?
9
5Probability
A ball is drawn at random from the box. What is
the probability it is green or black?
?
3
P(green or black)
?
9
6Probability
A ball is drawn at random from the box. What is
the probability it is green or black?
1
P(green or black)
3
7Geometric Probability
- Based on lengths of segments and areas of
figures. - Random
- Without plan or order. There is no bias.
8Probability 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
9Example 1
- Find the probability that a point chosen at
random on RS is on JK.
JK 3 RS 9 Probability 1/3
10Your Turn
- Find the probability that a point chosen at
random on AZ is on the indicated segment.
11Probability 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
12Example 2
- Find the probability that a randomly chosen point
in the figure lies in the shaded region.
8
8
13Example 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
14Example 3
- Find the probability that a randomly chosen point
in the figure lies in the shaded region.
5
15Example 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
16Your 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
17Solution
Find the area of the hexagon
?
6
?
?
3
18Solution
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
19Solution
19.57
Probability Shaded Area Total Area 19.57/113.1
0.173 17.3
6
113.1
3
20Example 4
- If 20 darts are randomly thrown at the target,
how many would be expected to hit the red zone?
10
21Example 4 Solution
Radius of small circles 5 Area of one small
circle 25? Area of 5 small circles 125?
10
22Example 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
23Example 4 Solution continued
Red Area100? Total Area 225? Probability
10
This is the probability for each dart.
24Example 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.
25Your Turn
- 500 points are randomly selected in the figure.
How many would likely be in the green area?
26Solution
- 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
27Solution
- 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
28Solution
- 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
29Solution
- 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.
30Summary
- 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.