Slide 2 / 66 Geometry

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Geometry

Probability

www.njctl.org

2014-09-08

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Table of Contentsclick on the topic to go to that section

· Probability of Simple Events

· Probability and Length

· Probability and Area

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Probability of Simple Events

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If the outcomes in a sample space are equally likely to occur, the theoretical probability of an event P(event) is a numerical

value from 0 to 1 that measures the likelihood of an event. You can write the probability of an event as a ratio, decimal or a percent.

P(event) = number of favorable outcomes number of possible outcomes

· An event with a probability close to 0 is unlikely to occur.· An event with a probability close to 1 is likely to occur.

· An event with a probability of 0.5 is just as likely to occur as not.

Impossible CertainEqually Likely to Occur

or not Occur

0 less likely 0.5 more likely 1

Theoretical Probability

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There are 7 red marbles and 3 green marbles in a bag. One marble is chosen at random.

Write the probability that a green marble is chosen.

P(Green)

Write as a fraction Write as a decimal Write as a percent

Non - Geometric Examples

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A sample space is a set of ALL possible outcomes for an activity or experiment.

A sample space is usually denoted using set notation {...} and the possible outcomes are listed as elements in the set

{a, b, c, ... z}.

Sample Space

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P(card) = 1/ 52

Suppose you choose a card from the deck. What is ....

P(Heart) = ______ P(3) = ______ P(4 of Spades) = ______

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Find the sample space in the box below each activity.

{yellow, blue, red, green}

{heads, tails}

{1, 2, 3, 4, 5, 6}

{yellow, green, red}

{(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)(4,1) (4,2)(4,3)(4,4) (4,5)(4,6)(5,1) (5,2)(5,3) (5,4) (5,5)(5,6)(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)}

Sample Space

Click to Reveal Click to

RevealClick to

RevealClick to

RevealClick to

Reveal

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P(2) = _______

P(even) = _______

P(prime) = _______

P(>4) = _______

Probability

2. Find the probability of each event.

1. Find the sample space for the activity below.

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P(green) = _______

P(orange) = _______


Probability

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P(heads) = _______

P(tails) = _______

2. Find the probability of each event.


Probability

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1 A multiple choice question has 14 possible answers, only one of which is correct. Is it "unlikely" to answer a question correctly if a random guess is made?

Yes

No

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2 What is the sample space for flipping a coin twice?

A HT TH

B HH HT TH TT

C HH HT TT

D HH TT HT HT

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3 What is the sample space for flipping a coin 3 times?

A HHH TTT THT HTH HHT TTH HTH

B HHH HTT HTH TTT HTT THH HHT THT

C HTT THT HTH HHH TTH TTT

D HHH HHT HTH HTT THH THT TTH TTT

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4 On a multiple choice test, each question has 4 possible answers. If you make a random guess on the first question, what is the probability that you are correct? A 4

B 1

C 1/4

D 0

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5 A die with 12 sides is rolled. What is the probability of rolling a number less than 11? A 1/12

B 10

C 5/6

D 11/12

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6 What is the probability of rolling a number greater than 2 on a number cube? A 1/6

B 1/3

C 1/2

D 2/3

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7 What is the probability of randomly choosing a science book from a shelf that holds 3 mystery books, 5 science books and 4 nature books?

A 1/4

B 1/3

C 5/12

D 7/12

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8 A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is blue?A 1/3

B 3/16

C 1/13 D 1/7

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We have evaluated probabilities by counting thenumber of favorable outcomes and dividing that number by the total number of possible outcomes.

In the rest of this unit, you will use a related process in which the division involves geometric measures such as length or area. This process is called geometric probability.

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Probability and Length


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Using Segments to Find Probability

A B C D

Point K on AD is chosen at random. The probability that K is on BC is the ratio of the length of BC to the length of AD.

P(K on BC) = BC AD

Fill in the blanks.

P(K on AC) = P(K on AB) = AD

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2 3 4 5 6 7 8 9 10 11 12 13 14

S Q R T

Point H on ST is selected at random. What is the probability that H lies on SR?

Step 1: Find the length of each segment.

length of SR = ____________ length of ST = ____________


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2 3 4 5 6 7 8 9 10 11 12 13 14

S Q R T

Point H on ST is selected at random. What is the probability that H lies on SR?

Step 2: Find the probability.

P(H on SR) = _______________

The probability is _______ or _________% .


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A B C D E F G H I J K L M

0 1 2 3 4 5 6 7 8 9 10 11 12

JL = __________

A point on AM is chosen at random. Find the probability that the point lies on the given segment.

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0 1 2 3 4 5 6 7 8 9 10 11 12

DJ = __________


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0 1 2 3 4 5 6 7 8 9 10 11 12

BE = __________


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0 1 2 3 4 5 6 7 8 9 10 11 12

AJ = __________


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0 1 2 3 4 5 6 7 8 9 10 11 12


CK = __________

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0 1 2 3 4 5 6 7 8 9 10 11 12


BL = __________

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In the figure at the right AB = 1 . BC 2

What is the probability that a point chosen at random on AC will lie on BC? Explain.

A B C


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A point between A and B on each number line is chosen at random. What is the probability that the point is between C and D?

A)

D)C)

B)

0 1 2 3 4 5 6 7 8

A C D B

0 1 2 3 4 5 6 7 8

A C D B

0 1 2 3 4 5 6 7 8

A C D B

0 1 2 3 4 5 6 7 8

A C D B


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Real-World Example

A commuter train runs every 25 minutes. If a commuter arrives at the station at a random time, what is the probability that the commuter will have to wait no more than 5 minutes for the train?

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Real-World ExampleWhich diagram models the situation? (Each number on the number line represents the number of minutes remaining before the next train leave)

0 5 10 15 20 25

D C E

0 5 10 15 20 25

D C E

0 5 10 15 20 25

D C EA) B) C)

Find the probability.

P(waiting no more = _______ than 5 minutes)

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A fitness club set up an express exercise circuit. To warm up, a person works out on weight machines for 90 s. Next, the person jogs in place for 60 s, and then takes 30 s to do aerobics. After this, the cycle repeats. If you enter the express exercise circuit at a random time, what is the probability that a friend of yours is jogging in place? What is the probability that your friend will be on the weight machines?

Real-World Example

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At the space museum, a movie starts every 15 min. There are 5 min between shows. If you enter the theater at a random time, what is the probability that you will have to wait more than 2 min for the next movie to start?

Real-World Example

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Real-World ExampleA Sunday night sports show is on from 10:00 p.m. to 10:30 p.m. You want to find out if your favorite team won last weekend but forgot that the show had already started. You turn it on at 10:14 p.m. The score is announced at one random time during the show. What is the probability that you haven't missed the repost about your favorite team?

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9 Point X on QT is chosen at random. What is the probability that X is on ST?A QT

ST

B STQT

C QSST

DSTQS

Q R S T

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10 What is the probability that a point chosen at random from EH will be on EF?

A 1/3

B 3

C 1/4

D 3/4

3 cm 5 cm 4cm

E F G H

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11 If AC = 10, what is the probability that a point chosen at random from AC will land on BC?

A 3/5

B 2/5

C 2/3

D 1/2

A B C

6 in

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12 Find the probability that a point chosen at random on AE is on BD.

A 20%

B 25%

C 30%

D 35%

E 40%

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13 Point P on AD is chosen at random. For which of the figures below is the probability that P is on BC 25%? Note: Diagrams not drawn to scale.

A

B

C

D

A B C D

2 5 8 10

A B C D

2 3 4 5

A B C D

1 2 3 4

A B C D

1 2 3 5

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14 You have a 7-cm straw and a 10-cm straw. You want to cut the 10-cm straw into two pieces so that the three pieces make a triangle. If you cut the straw at a random point, what is the probability that you can make a triangle?

A 30%

B 40%

C 60%

D 70%

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Probability and Area


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A

C

Using Area to Find ProbabilityPoint B in region A is chosen at random. The probability that point B is in region C is the ratio of the area of region C to the ratio of the area of region A.

P(B in region C) = area of region C area of region A

Find the probability for the given areas.

area of region A = 24 in2 area of region C = 3 in2

P(B in C) = ________

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4 in.

A triangle is inscribed in a square. Point N in the square is selected at random. What is the probability that N lies in the shaded region.

Step 1: Find the area of each region.

area of shaded region = _________ area of square = __________

Step 2: Find the probability.

P(N is in shaded region) = _______________

The probability is _______ or _________% .

Using Area to Find Probability

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7

16

4

4

Find the probability that a point chosen at random in the trapezoid with a height of 4 will lie in either of the shaded regions.


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16

10

A point in the figure to the right is chosen at random. Find the probability to the nearest percent that the point lies in the shaded region.


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A point in the figure to the right is chosen at random. Find the probability to the nearest percent that the point lies in the shaded region. 7 mm

10 mm


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A dart is thrown at random at this dart board. If the dart hits the board, find the probability to the nearest percent that it will land in the shaded region.

12 ft

6 ft


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30 cm

50o

A point in the figure to the right is chosen at random. Find the probability to the nearest percent that the point lies in the shaded region.


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Assume that a dart you throwwill land on the 1-ft square dartboard and isequally likely to land at any point on theboard. Find the probability of hitting each ofthe blue, yellow, and red regions. The radii ofthe concentric circles are 1, 2, and 3 inches,

Using Area to Find Probability - Concentric Circles

P(blue) = P(yellow) = P(red) =

12 in

12 in

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A point in the figure is chosen at random. Find the probability that the point lies in the shaded region.

3 cm

4 cm2 cm

Using Area to Find Probability - Concentric Circles

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A dart is thrown at random at the dart board to the right. If the dart hits the board, find the probability to the nearest percent that it will land in the shaded region.

If dimensions aren’t given, CHOOSE YOUR OWNA good number to use is, An Even Number

HINT:


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This is a square with four semi-circles. HINT:


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In the fundraiser game at the right, players toss darts at a board to try to get them into one of the holes. The diameter of the center hole is 8 in. The diameter of each of the four corner holes is 5 in. The board is a 20-in.-by-30-in. rectangle. Find the probability that a tossed dart will go through the indicated hole.

WIN

Dart Toss

a.) center hole

b) any corner

c) top right or left

d) bottom left


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15 If a dart hits the target at random, what it the probability that it will land in the shaded region?

A 1/3

B 7/16

C 1/9

D 1/4

2 in

6 in

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16 Find the probability that an object falling randomly on the figure will land in the shaded area.

A 0.32

B 0.36

C 0.50

D 0.26

20 in

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17 What is the probability that a randomly dropped marker will fall in the non-shaded region?

A 1/16

B 1/4

C 15/16

D 4

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18 Two concentric circles have radii of 11 cm and 17 cm. Find the probability to the nearest thousandth that a point chosen at random from the circles is located outside the smaller circle and inside the larger one. A 0.021B 0.097C 0.581D 0.647

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19 Find the probability that a point chosen at random in the regular triangle lands in the shaded region.

A 25 %

B 30 %

C 33.3 %

D 40 %

3

6

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8

4

20 Find the probability that a point chosen at random lands in the shaded region. Round to the nearest tenth, if necessary.

A 39.3 %

B 60.7 %

C 64 %

D 36 %

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21 Find the probability that a point chosen at random in the circle lands in the shaded region. Round to the nearest tenth.

A 6.9%

B 26.8 %

C 50.0%

D 55.6%

E 27.8%

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