Example 6

A cricket ball is drawn at random from a bag containing 9 red balls, 5 white balls and6 yellow balls. Calc~late the probability of drawing(a) a red or a white ball (b) a white or a yellow ball.

Solution................•-0:>:loOJ»~r-~-lImo~

(a) The number of red balls, n(R) = 9The number of white balls, n(W) = 5The total number of balls, n(S) = 9 + 5 + 6 = 20.. P(Ru W) = P(R)+ P(W)

= n(R) + n(W')n(S) n(S)9 5= 20 + 2014= 207= 10

Hence, the probability of drawing a red or a white ball is 170'

(b) The number of white balls, n(W) = 5The number of yellow balls, n(y) = 6. . P(W u Y) = P(W) + P(Y)

= n(W) + n(Y)n(S) n(S)5 6= 20 + 2011= 20

Hence, the probability of drawing a white or a yellow ball is 16'Exercise 13.4

1. A golf ball is drawn at random from abag containing 15 white balls, 12 redballs and 13 orange balls. Calculate theprobability of drawing either(a) a white or an orange ball(b) a red or an orange ball

2. In a class of 40 students who either likeCalypso or Reggae but not both; 25 likeCalypso and 15 like Reggae. If a student ischosen at random from this class, what is

the probability that the student likes eitherCalypso or Reggae?

3. In a class of 40 students who only swim oronly ride; 23 swim and 9 ride. If a studentis chosen at random from this class,calculate the probability that the studenteither swims or rides.

4. On a rainy day, of the 36 students in aclass; 12 wore a hat and 15 wore a jacket.No student wore both a hat and cap.

""U;;0oOJ»!:!:!,.~--lImo~

Calculate the probability that a studentchosen at random from the class woreeither a hat or a jacket.

5. There are 124 books in a school libraryof which 15 are written in Spanish and17 are written in French. A student picks abook at random from the library. Calculatethe probability that the book is written ineither Spanish or French.

6. Let A and B be any two mutually exclusiveevents with P(A) =iand P(B) = t. FindP(A u B).

7. Let A and B be any two mutuallyexclusive events with P(A)= t andP(B) = i- Calculate P(A u B).

8. X and Yare two mutually exclusiveevents with P(X) = 1~ and P(X u Y) = lEvaluate PO,).

9. Rand Q are two mutually exclusiveevents with P(Q) = ~ and P(R u Q) = t.Find P(R).

10. Rand 5 are two mutually exclusiveevents with P(R) = 130 and P(S) =~.Calculate P(R uS).

, _-r.~~~ ~ ~-~r

.."-~~ I!-- -,' -,::- ~~ ~I ~ ;_~_

nditional Probability and Dependent Events

If A and B are any two events, from the same or different experiments, then the conditionalprobability that A occurs given that B has already occurred, is written as P(A I B).

Conditional probability is the probability of an event occurring given that another event hasalready occurred.

s--------------------------~

B

Consider the Venn diagrams shown above.The first Venn diagram represents the sample space 5 and two events A and B.

h n(B)T us P(B)=n(S)'

After the event B has occurred, then the sample space 5 is reduced to a sample space B, asshown in the second Venn diagram.

Thus P(A I B) = n(~(~) B)

""UAIoOJl>~r-~--iImo~

Solution................•

R W n5 7 125 6 11

(a) P(W2 I W1) = 161

(b) P(R I W) = 151

Example 9

A card is chosen at random from a pack of 52 playing cards. A second card is chosen withreplacing the first card. What is the probability that the first card chosen is a queen and thesecond card chosen is a king?

P(Q) = 5iAnd P(K I Q) = 541

So . P(Q n K) = P(Q) . P(K I Q)4 4=_.-52 514= 663

Hence, the probability is 6i3' that the first card chosen is a queen and the second cardchosen is a king.

Solution................•

Now

Exercise 13.5

1. A die is thrown and an odd number turnsup. What is the probability that it is aprime number?

2. A bag contains 8 blue marbles and6 green marbles. A blue marble isremoved and not replaced. Calculatethe probability that the second ballremoved is(a) blue (b) green

3. A die is thrown twice. Find the probabilityof getting a number greater than threein the second throw given that an evennumber was obtained in the first throw.

4. A jar contains red and green marbles. -marbles are chosen from the jar and n -replaced. The probability of selecting ared marble then a green marble is 0.35and the probability of selecting a redmarble on the first draw is 0.57. Whatsthe probability of selecting a green rna:on the second draw, given that the firstmarble drawn was red?

5. At a high school, the probability thastudent takes Information Technologvand Spanish is 0.45. The probability -a student takes Information Technol _

is 0.75. Calculate the probabilitythat a student takes Span ish giventhat the student is taking InformationTechnology.

6. In a school, the probability that astudent owns a computer is 0.72 andthe probability that a student owns acomputer and a flash drive is 0.45.Evaluate the probability that a studentowns a flash drive, given that the studentowns a computer.

7. In a class, the probability that a studentplays cricket and football is 0.54. Theprobability that a student plays football is0.86. Find the probability that a studentplays cricket given that the student playsfootball.

8. In a Caribbean country the probabilitythat a household has a television is 0.98and the probability that a householdhas a television and a computer is 0.64.Calculate the probability that thehousehold has a computer given that ithas a television.

9. A card is drawn from a pack of 52playing cards and it is a heart. The cardis not replaced. A second card is drawn.Calculate the second card is also a heart.

10. A class wrote two Mathematics tests.82% of the class passed both tests and94% of the class passed the first test.A student is chosen at random from this

class. What is the probabi Iity that thestudent passed the second test given thatthe student passed the first test?

11. A class has 24 girls and 16 boys.A teacher chooses one student to answera question; then he chooses anotherstudent. What is the probabi Iity that bothstudents are(a) boys? (b) girls?

12. In a shipment of 60 computers, 5 weredamaged. Two computers were chosenat random, checked and not replaced.What is the probability that bothcomputers were damaged?

13. On a Mathematics test, 12 out of40 student got an A grade. Twostudents are chosen at random withoutreplacement; what is the probability thatboth students got an A grade on the test?

14. In a survey, it was found that 9 out of40 students in a class took a bus toschool. If two students are selected atrandom without replacement, what is theprobability that both students took a busto school?

15. A box contains 12 good bulbs and2 defective bulbs, which are mixedup. A bulb is tested and not replaced.A second bulb is tested and not replaced.Calculate the probability that both bulbswere found.

-0;:0oOJ»~,~--IImo~

The event B = {(H, 6), (T, 6)}

The event C = {(H, 6)}

Thus P(A) = n(A) ::c ~ = 1n(S) 12 2

P(B) = n(B) = l = 1n(S) 12 6

n(C) 1P(C) = n(S) = 12 = P(A n B)

P(A n B) = P(A) . P(B) = t x t = 112

P(q = P(A n B) = 112

, then events A and B are independent.

The possibility space diagram is shown below.

-0:>J0

andOJ»~r-~ Also-lIm0 Since;;0-<

570

H

• •• •

T

• • ••• •• •

1 2 3 4 6

Exercise 13.6

1. A jar contains 4 red, 7 green and 5 yellowmarbles. A marble is chosen at randomfrom the jar with replacement. A secondmarble is then chosen. What is theprobability of choosing a red and a yellowmarble?

2. A school survey found that 7 out of10 students like gyros. If two studentsare chosen at random with replacement,what is the probabi I ity that both studentslike gyros?

3. A spinner with nine equal sectorsnumbered 1 to 9 is spun, and a fair coin

5

is tossed. Calculate the probability ofgetting an even number on the spinnerand a tail on the coin.

4. A survey showed that 67% of all thestudents in a school dislike eatingvegetables. If two students are chosen atrandom, what is the probability that bothstudents dislike eating vegetables?

5. A red die and a blue die are rolledtogether. What is the probability that two5 s occur?

6. The probability that a man in Japan will bealive in 10 years is 0.8 and the probabilitythat a woman in Africa will be alive in10 years is 0.4. Calculate the probabilitythat they will both be alive in 10 years.

7. A fair die is tossed twice. Find theprobabi Iity of getti ng a 1, 3 or 5 on thefirst toss and a 2, 4 or 6 in the second toss.

8. Two pens are drawn successively withreplacement from a box which contains5 black and 4 green pens. Find theprobability that(a) the first pen is black and the second

is green(b) both pens are black.

9. Two letters are picked from the wordMATHEMATICS with replacement. Findthe probability that(a) the first letter is E and the second

letter is 5

(b) the first letter is a M and the secondletter is a A

(c) the first letter is H and the secondletter is a M.

10. The 5 vowels in the English alphabetwere written on cards and placed in abag. The 21 consonants in the Englishalphabet were written or cards andplaced in another bag. Yuri picked onecard from each bag at random. Find theprobability that he picked the letters 'Y'and 'U'.

ity Space Diagram, Venn Diagrame Diagram

Possibility Space DiagramA possibility (probability or sample) space diagram can be used to solve a problem involvingprobability.

Example 13

Two fair dice, a blue die and an orange die, are rolled simultaneously.(a) Construct a possibility space diagram to represent all the possible outcomes.(b) Calculate the probability that the sum of the dots in a single roll of the two dice is

(i) always greater than 6. (ii) always less than 6.

Solution................•

(a) Operation Blue die

+ 1 2 3 4 5 61 2 3 4 5 6 7

0) 2 3 4 5 6 7 8-00) 3 4 5 6 7 8 9coC 4 5 6 7 8 9 10ro...0 5 6 7 8 9 10 11

6 7 8 9 10 11 12

The possibility space diagram is shown above.

'""C;;0oCD»~,~-IImo~

Hence, the probability of removing two discs of different colours is ~~.

(c) P(same colour) = P(R (J R) + P(C (J C)=.lQ+36

91 9146= 91

Hence, the probability of removing two discs with the same colour is ~~.

Exercise 13.7

1. A circular spinner is subdivided into fiveequal sectors and a triangular spinner issubdivided into three congruent isoscelestriangles as shown below.

Spin the pointer of the circular spinnerand the triangular spinner simultaneously.

(a) Complete the sample space diagramgiven below.

Triangular spinner

'- X 2 3 5iJ.)c 1c0...

1tJl

'-(1j 4::JU 4'-.-u 6

(b) Calculate the probability that theproduct of the numbers on the two

. .spinners IS

(i) less than 10(ii) greater than 10

2. Two circular spinners with f equal sectorseach are shown below.

Each pointer is spun once.

(a) Construct a possibility space diagramto represent all possible outcomes.

(b) Find the probability that(i) both pointers stop at the same

number(ii) the pointer of the orange spinner

shows the larger number.

3. Two fair dice are thrown together.(a) Construct a probabi Iity space

diagram to represent all possibleoutcomes.

(b) Calculate the probability that(i) both dies turn up the same

number of dots(ii) the first die turns up the greater

number of dots.

4. A and B are two independent events withP(A) = 0.2 and P(B) = 0.3.

(a) Find(i) P(A (l B) (ii) P(A (l B') (iii) P(B (l A')

(iv) P(A uB) (v) P(A uB)'(b) Draw a Venn diagram to represent the

probabi Iities.(c) Calculate P(S).

5. Rand Q are two independent events withP(R) = 0.25 and P(Q) = 0.5.(a) Evaluate

(i) P(R (l Q) (ii) P(R (l Q') (iii) P(Q (l R')(iv) P(RuQ) (v) P(RuQ)'

(b) Draw a Venn diagram to represent theprobabilities.

(c) Calculate P(S).

6. X and Yare two independent events withP(X) = i and P(Y) = ~.(a) Determine

(i) P(X (l Y) (ii) P(X (l Y') (iii) P(Y (l X)(iv) P(X u Y) (v) P(X uY)'

(b) Draw a Venn diagram to represent theprobabilities.

(c) Calculate P(S).

7. A lottery machine contains 4 purple ballsand 8 yellow balls. The balls are mixedby the machine and a ball released. Thecolour of the ball is noted and replaced.The lottery machine repeats the processwith a second ball being released.(a) Construct a tree diagram to show

all the possible outcomes of theexperiment.

(b) Calculate the probability that(i) 2 purple balls were released

(ii) at least one yellow ball wasreleased.

8. A pick two machine contains 6 orangeballs and 8 white balls. The balls are

mixed by the machine and a ballreleased. The colour of the ball is notedand it is not replaced. The pick twomachine repeats the process with asecond ball being released.(a) Construct a tree diagram tree to show

all the possible outcomes of theexperiment.

(b) Calculate the probability that(i) 2 balls were released

(ii) at least one orange ball wasreleased.

9. A pick two machine contains 3 blueballs and 5 yellow balls. A lotto machinecontains 6 blue balls and 8 yellow balls.The balls are mixed in the machines.A ball is released from each machineand its colour is noted.(a) Draw a probability tree to show

all the possible outcomes of theexperiment.

(b) Find the probability that(i) both balls that were released are

blue(ii) one blue and one yellow ball

were released.

10. A trick die has 6 dots each on four sidesand 5 dots each on the other two faces.

A fair silver dollar is flipped and the trickdie is rolled.(a) Construct a probability tree to show

all the possible outcomes of theexperi ment.

(b) Find the probability that(i) a head and a 6 turn up

(ii) a tail and a 5 turn up.

Exercise 13.31. P(Q u H) = 1i2. P(K u 0) = 172

3. P(T u C) = !~4. P(M u P) = 1

~ 5. P(C u 5) = ~~

. ~ 6. P(C u B) = i~7. P(S u0) = t8. P(A n B) = 152

9. P(A n B) = i~10. P(X u Y) = i~Exercise 13.4

10 1. (a) P(Wu 0) = 170

(b) P(R u0) =~2. P(CuR)=1

3. P(S u R) =~

4. P(H uJ) = %5. P(S u F) = 381

6. P(A uB) = ~~87. P(A u 8) =g

238. P(Y) = 88

9. P(R)= 2~

10. P(Ru 5) = 170

Exercise 13.51. P(prime nurnberlodd number) = t2. (a) P(B2IB')=173

(b) P(ClB) = 163

3. P(AIB) = t·4. P(CIR) = 0.6145. p(SII) = 0.66. p(FIC) = 0.6257. P(CIF) = 0.628

8. p(CI n = 0.653

9. P(H2IH,)= ~~10. p(T

2IT,) =0.872

11. (a) P(B,n B2)= 123

(b) P(C, n C2) = ~~

12. P(O, n 02) = 1~7

13. P(C, n C2) = 11310

14. P(B, n B2) = is15. P(O, n 02) = 9\

Exercise 13.6

1. P(R n Y) = :4

2. P(C n C) = 1~0

3. prE n T) = ~

P( ) 44894. On ° = 10000

5. P(5 n 5) = 316

6. P(Jn A) = 0.32

7. pro n E) =t8. (a) P(B n C) = ~~

(b) P(B n B) = ~~

9. (a) prE n 5) = 1i1

(b) P(M n A) = 1i1(c) P(H n M) = 1~1

10. pry n U) = 1bs

Exercise 13.71. (a) Triangular spinner

X 2 3 5•....

1 2 3 5Q)cc 1 2 3 50..tfl

4 8 12 20•....ro::J 4 8 12 20u•.....- 6 12 18 30u

(b) (i) P(product less than 10) = 185

(ii) P(product greater than 10) = 175

2. (a) Orange spinner1 2 3 4 5

•.... 1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5)Q)c 2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5)c0..

3 (3,1) (3, 2) (3, 3) (3, 4) (3, 5)tfl

...:.::4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5)c

CL5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5)

or

5 0 0 0 0 0

~ 4 0 0 0 0/ 0cc.~ 3 0 0 /0 0 0

~ /Cii: 2 0 0 0 0 0

1 r6/0 0 0 0

2 3 4 5Orange spinner

(b) (i) P(both spinners stop at the samenumber) = t

(ii) P(orange spinner shows thelarger number) =i

3. (a)6 0 0 0 0

5 0

(1);.04 0 0"0C8 3 0 0<U

U'1

2 0 0

1 0/0»

0 0 0 0 zVl

~m;u

2 3 4 5 6 Vl

First die

(b) (i) P(both dies turn up the samenumber of dots) = t

(ii) P(first die turns up the greaternumber of dots) = 152

4. (a) (i) P(A n B) = 0.06

(ii) P(A n B') = 0.14

(iii) P(B n A') = 0.24

(iv) P(A u B) = 0.44

(v) P(A u B)' = 0.56(b) P(S)------------.

(c) P(S) = 1

5. (a) (i) P(R n Q) = 0.125

(ii) P(R n Q') = 0.125

(iii) P(Q n R') = 0.375

(iv) P(R u Q) = 0.625

(v) P(R u C)' = 0.375

»zVl::z:m;;0Vl

(b) (i) P(Pun P) = ~(ii) P(atleast one yellow ball) = P(Pun Y) + P(Y n P) + P(Y n Y) = ~

8. (a) First release Second release Outcome Probability3 5 15

P(O nO) =""7 x 13 = 9T

(b) peS)

(c) peS) = 16. (a) (i) P(X n Y) = *

(ii) P(X n Y') = }o(OO.) P(Y X') 3III n ="8

7. (a) First release Second release

1"3

LPu 2 Y

"3

y~PU

Gy

3""7

yoCo 8 W

13

Outcome

PunPu

Pun Y

YnPu

YnY

OnO

OnW

WnO

WnW

(iv) P(X u Y) = !6(v) P(X u Y)' = :0

(b) P(S)-----------.

(c) peS) = 1

Probability1 1 I

P(Pu n Pu) ="3 x "3 = "9

2 I 2P(Y n Pu) = "3 x "3 = "9

2 2 4P(Y n Y) ="3 x "3 = "9

3 8 24P(O n W) =""7 x 13 = 9T

4 6 24peW n0) =""7 x 13 = 9T

474pew r, W) = ""7 x 13 = 13

(b) (i) P(W (1 \IV) = 1~ 1 ,

(ii) P(atleast one orange ball) = P(O (10) + P(O (1W) + P(W (10) = ?39. (a) Pick two lotto Outcome Probability

3 3 9PCBn B) = 8 x ...", = 56

(b) (i) P(B (1 B) = 596(ii) P(one blue and one yellow) = P(B (1 y) + P(Y (1 B) = ~~

10. (a) Coin Die Outcome Probability1 2 1

P(H n 6) = "2 x "3 = "3

B

58

1"2

T/6~S

3

(b) (i) P(H (16) = t(ii) P(T (15) = t

Chapter 14Kinematics of Motion Alonga Straight LineExercise14.11. (a) _tf--__ -_7 _m~~~~_=C+ S m--; •

A 0 A"'II -- 7 m ---+.+-- S m ---.'II 12m •

(b) AA = 12 m

Hn6

HnS

Tn6

TnS

BnB

BnY PCBn Y) = 2 x .±. = 28 7 14 »z

If)

::Em;0If)

YnB P(Y n B) = 1.- x 2 = Q8 7 56

YnY 5 4 5P(Y n Y) = 8 x ...", = 14

713

P(H n S) = 1.. x 1.. = 1..236

peT n 6) = 2 x 1.. = 1..3 2 3

2. (a) AC = 11 m

(b) (i) OA = +3.5 m

(ii) OB = -2 m(iii) OC = -7.5 m

3. 5 = 10.4 m/s4. 5 = 22.8 km/h5. (a) v = 2.92 m/s northeast

(b) v = 10.5 km/h northeast