Geometric Probability 8 0 - iMater

08

Geometric Probability

MathemaHcs Florida Standards

Prepares for MAFS.912.S-CP.1.1 Describe events as subsets of a sample space...using characteristics... of the outcomes, or as unions, intersections, or complements of other events.,..

MR 1, MP 3. MP 4

Objective To use segmentand area modelsto find the probabilities ofevents

fi "1 Getting Ready!

X C

Try making a chart of all the possible outcomes to make sense of this problem.

A fair coin is equally likely to land heads up or tails up.

Suppose you toss a fair coin three times. What is the probability that the coin will land toils up exactly twice? Explain your reasoning.

PRA^Ic^ES Solve It, you found a probability involving a coin.In this lesson you willfind

probabilities based on lengths and areas.The probability ofan event,written P(event),

is the likelihood that the event will occur.

When the possible outcomes are equally likely,the theoretical probability ofan

event is the ratio ofthe number offavorable outcomes to the number of

possible outcomes.

Lesson

^

Vocabulary

? geometric

probablilty

P(event) nnuummbbeerroofffpaovsosriabblleeoouuttccoommeess

Recall that a probability can be expressed as a fraction, a decimal, or a percent.

Essential Understanding You can use geometric models to solve certain types

of probability problems.

In geometric probability,points on a segment or in a region ofa plane represent outcomes.The geometric probability ofan eventis a ratio that involves geometric measures such as length or area.

Key Concept Probability and Length

PointSon AD is chosen at random.The probability thatS is

D

on BC is the ratio ofthe length of BC to the length of AD.

P(Son^)=|?

668 Chapter 10 Area

How can you find the length of each segment?

You can use the Ruler Postulate to find the

length of each segment.

Problem 1 Using Segments to Find Probability

Point iCon ST is chosen at random.

What is the probabilitythatKlies on QRl

--, length of QR |5 P(^K on QR)= ,

length ofSr 2- 14

S

0

-1 1 h

H

2 34

3 1

12' ^^4

h

H h

8 9 10 11 12 13 14

The probabilitythatKis on QR is or 25%.

il;^ Got It? 1.Usethe diagram in Problem 1.PointHon STisselected atrandom.Whatis

the probability thatHlies on SRI

How can you draw a diagram to model the

situation?

Draw a line segment with endpoints 0 and 25 to represent the length of time between trains. Each point on the

segment represents an arrival time.

Problem 2 Using Segments to Find Probability

Transportation A commuter train runs every25 min.Ifa commuter arrives at the station at a random time,what is the probability that the commuter will have to wait

at least 10 min for the train?

Assume that a stop takes very little time. Draw a line segment to model the situation.The length ofthe entire segment represents the amount

of time between trains. A commuter will have to wait at least 10 min for

the train ifthe commuter arrives at anytime between 0 and 15 min.

P(.waiting atleast 10 mm)- = leinegt,,lgi,ohfoffaevnotriarbelseesgemgemnetnt = 2155' 53

H

h

0 5 10 15 20 25

4--10 5 10 15 20 25

The probability that a commuter will have to wait at least 10 min for the train

is|,or60%.

Got It? 2. Whatis the probability that a commuter will have to wait no more than

5 min for the train?

When the points ofa region represent equally likely outcomes,you can find probabilities by comparing areas.

Key Concept Probability and Area

Point Sin region R is chosen at random.The probability thatS

is in region Nis the ratio ofthe area ofregion N to the area of

region R.

area ofregion N

P(Sin region =



Lesson 10-8 Geometric Probability

J 669

Using Area to Find Probability

A circle is inscribed in a square.Point Q in the square is chosen at random.Whatis the probability that Q lies in the shaded region?

6 cm

The length of a side of the square, which is also the length of the diameter of the inscribed circle

The areas of the square and the shaded region

Subtract the area of the circle from the area of the

square to find the area of the shaded region.Then use it to find the probability.

area ofshaded region = area ofsquare - area ofcircle

= 6^-77(3)2

= 36 - 977

area ofshaded region

P{Q lies in shaded region)=

area ofsquare

^-0.215

The probability that Qlies in the shaded region is about0.215,or 21.5%.

Got It? 3. A triangle isinscribed in a square.Point Tin the square is

selected at random.Whatis the probability that Tlies in the shaded region?

How can you find the

area of the red zone? The red zone lies between two concentric circles. To find the area

of the red zone,subtract the areas of the two

concentric circles.

Problem 4 Using Area to Find Probability

Archery An archery target has5 colored scoringzonesformed by concentric circles.The target's diameter is 122cm.The radius ofthe yellow zone is 12.2cm.The width ofeach ofthe other zonesis also 12.2cm.Ifan arrow hits the target at a random point, what is the probability that it hits the red zone?

The red zone is the region between a circle with radius 12.2 + 12.2, or 24.4cm and the yellow circle with radius 12.2 cm.The targetis a circle with radius or 61 cm.

P(arrow hits red zone)= area ofred zone area of entire target

77(24.4)2 _ 7r(12.2)2

77(61)2

= 0.12

The probability ofan arrow hitting a point in the red zone is 0.12,or 12%

Got It? 4. a. What is the probability that an arrow hits the yellow zone? b. Reasoning Ifan arrow hits the target at a random point,is it more likely

to hit the black zone or the red zone? Explain.

670 Chapter 10 Area

Lesson Check

Do you know HOW?

Point Ton AD is chosen at random.Whatis the

probability that Tlies on the given segment?

H h

C D H h

8 9 10

1. AB

2. AC

3. BD

4. BC

5. A point Kin the regular hexagon is chosen at random. Whatis the probability thatKlies in the region thatis

not shaded?

18 cm 10.4 cm

,,MATHEMATICAL

Do you UNDERSTAND? PRACTICES

6. Reasoning In the figure

5 Q

the right,^=2- Whatis ' *

the probability that a point on ST chosen at random will lie on QT7 Explain.

7. Error Analysis Your class needs to find the probability that a point A in the square chosen at random lies in the shaded region.Your

classmate's work is shown below.

What is the error? Explain.

shaded

Area of semj^ipctgs square

Practice and Problem-Solving Exercises CQJpraoIce^^

Practice

A point on AK is chosen atrandom.Find the probabilitythatthe pointlies on the given segment.

A B CD E F GH I J K

0-H 1 h

1 hH ^ 1 #

8 9 10

^ See Probiem 1.

8. CH 11. B

9. FG 12. AK

10. DJ 13.^

14. Transportation At a given bus stop,a city bus stops every 16min.Ifa student arrives at his bus stop at a random time,whatis the probability that he will not

have to wait more than 4 min for the bus?

^ See Problem 2.

15. Traffic Lights The cycle ofthe traffic light on Main Street at the intersection of Main Street and Commercial Street is 40 seconds green,5seconds yellow,and 30 seconds red.Ifyou reach the intersection at a random time,whatis the probability that the light is red?

16. Communication Your friend is supposed to call you between 3 p.m. and 4 p.m. At3:20 P.M., you realize that your cell phone is off and you immediately turn it on. What is the probability that you missed your friend's call?

L I Lesson 10-8 Geometric Probability

671

A pointin the figure is chosen atrandom.Find the probability thatthe point lies in the shaded region.

17.

18.

19. 4ft

=:-- 5 m

3 m

^ See Problems 3 and 4.

20.

:)c;

12 in.

Target Game A target with a diameter of14cm has4scoring zones formed by concentric circles.The diameter ofthe center circle is2cm. The width ofeach ring is 2cm.A dart hits the target at a random point. Find the probability that it will hit a point in the indicated region.

21. the center region

22. the blue region

^Apply

23. either the blue or red region 24. any region

25. Points M and Nare on ZB with M betweenZand N.ZM = 5, NB =9, and ZB = 20.A point on ZB is chosen atrandom.Whatis the probability thatthe point is on MAf?

26. BZ contains MN and BZ= 20.A point on BZ is chosen at random.The probability thatthe pointis also on MN is 0.3, or 30%.Find MN.

27. Think About a Plan Every 20 min from 4:00 p.m.to 7:00 p.m.,a commuter train crosses Boston Road.For3 min,a gate stops carsfrom crossing over the tracks as the train goes by. Wliat is the probability that a motorist randomly arriving at the train crossing during this time interval will have to stop for a train? ? How can you represent the situation visually? ? Wliat ratio can you use to solve the problem?

28. Reasoning Suppose a pointin the regular pentagon is chosen at random.Whatis the probability that the point is notin the shaded region? Explain.

29. Commuting A bus arrives at a stop every 16 min and waits 3 min before leaving. What is the probability that a person arriving at the bus stop at a random time has

to wait more than 10 min for a bus to leave?

025^30. Astronomy Meteorites(mostly dust-particle size)are continually bombarding

Earth.The surface area ofEarth is about65.7 million mi^.The area ofthe United

States is about3.7 million mi^.Whatis the probability that a meteorite landing on

Earth will land in the United States?

31. Reasoning What is the probability that a point chosen at random on the circumference of O C lies on AB ? Explain how you know.

32. Writing Describe a real-life situation in which you would use geometric probability.

672 Chapter 10 Area

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