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MATHEMATICS WORKSHEET
XI Grade (Semester 1)
Chapter 2
by: Ignatia Maria Midawati
SMAK ST. ALBERTUS
(ST. ALBERT Senior High School)
Talang 1 Street Malang 65112, Indonesia
Phone (0341) 564556, 581037 Fax.(0341) 552017
Email: [email protected] homepage: http://www.dempoku.com,
www.mathdempo.blogspot.com , www.mida39.wordpress.com
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PREFACE Mathematics Module “PROBABILITY” is written for students of St. Albert Senior High School at XI Grade Semester 1. The contents are arranged under some worksheets which the students can fill to learn about PROBABILITY. Each worksheet is expanded in detail and step by step manner for easy understanding. It starts with a brief introduction and explanation follwed by filled examples and numerous simple exercises to build up the student’s technical skills and to reinforce his or her understanding. It is hoped that this approach will enable the individual student working on his or her own to make effective use of the module besides enabling the teacher to use them with mixed ability groups. Finally, we would like to thank to all those involved in the production of this module.
Ignatia Maria Midawati
CONTENTS
Page Term in Mathematics - Probability .............................................................................. 3 Worksheet 1 – Factorial, The Multiplication and Basic Counting Principle.................. 4 Worksheet 2 – PERMUTATIONS ................................................................................ 9 Worksheet 3 – COMBINATIONS and BINOMIAL THEOREM................................. 16 Worksheet 4 – PROBABILITY .................................................................................... 28 Worksheet 5 – Independent and Dipendent Events ...................................................... 36 Worksheet 6 – Addition of Probabilities (Mutually Exclusive Events)........................ 46 Worksheet 7 – Summary and Problem Solving ............................................................ 54
First published August, 2008
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BIBLIOGRAPHY
1. Cambridge, IGCSE Mathematics, Karen Morrison 2. New Syllabus Mathematics 3, Teh Keng Seng BSc, Dip Ed & Looi Chin Keong BSc.
Dip Ed 3. New Syllabus Mathematics 4, Teh Keng Seng BSc, Dip Ed & Looi Chin Keong BSc.
Dip Ed
4. Longman, Pre-U Text STPM Mathematics S Paper 2, Soon Chin Loong, Tong Swee Foong, Lau Too Kya, Pearson Malaysia SDN. BHD., 2008.
5. New Additional Mathematics, Ho Soo Thong, Dip Ed, Khor Nyak Hiong, Bsc, Dip Ed
TERM in MATHEMATICS PROBABILITY
NO Mathematics
Expressions How to read in English
1. 5 + 2 = 7 Five plus two is equal to seven
2. 9 – 3 = 6 Nine minus three is equal to six
3. 7 x 9 = 63 Seven times nine is equal to sixty three
4. 12 : 3 = 4 Twelve devided by three is equal to four
5.
2
1 ,
4
1
a half , a quarter
6.
3
2 ,
5
3 , 2
3
1
Two thirds , three fifths , two and one third Two over three, three over five, two and one over three
7. (x+y)n (x plus y) power n
8 n! n factorial.
9. P(A) Probability of event A
10 P(A and B) Probability of event A and B
11. P(E2 | E1) Probability of event E second after E first
12. 25,731 Twenty five thousands and seven hundred thirty one
13. 25.731 Twenty five point seven three one
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Topic : FACTORIAL, THE Multiplication and BASIC COUNTING
PRINCIPLE TIME : 4 X 45 minutes
STANDARD COMPETENCY :
1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving.
BASIC COMPETENCY:
1.5 To use the rules of multiplication , permutation, and combination in problem solving. In this chapter, you will learn about : • How to do operation mathematics in Factorial • How to use The Multiplication and Basic Counting Principle A. Factorial (!)
1!0 =
1!1 =
12!2 ×=
123!3 ××=
KKKKKKKK ×××=!4 . :
KKKKKKKKKKK=!n
Hence we have :
The number of permutations of n different objects is 123...........................................! ×××=n Note: We may write !4 as 1234 ××× or !34× depends on the problems.
Worksheet 1Worksheet 1Worksheet 1Worksheet 1stststst
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Example 1 Simplify: 1. 5! = … x … x … x … x … = …
2. KK
KK =×=!9
!10
3. KKKKK
KKK =×=×
××=!2!9
!11
4. ( ) KKKKKKK =−=−×==− !9!10
5. K
K=10.11.12
6. ( )( ) K=++ !12 nnn
7. ( ) KK
KK =×=− !1
!
n
n
8. ( )( ) K
K
KKK =××=−+
!1
1
n
n
9. ( )( ) K
K
KK
K =×
=−−
!1
!2
n
n
10. If ( ) 20!2
! =−n
n, find the value of n .
11. KKK=!3!2
!5
12. ( ) KKK=− !210!2
!10
13. ( ) KKK=− !3!3
!
m
m
B. The Multiplication and Basic Counting Principle
Example 2 There are 2 shirts and 3 pants. Ryan will use for suits of his clothes. How many possible suits can he use?
Solution
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We can use a tree diagram to systematically list the various possibilities as shown below. Example 3 There are 4 ways from Malang to Surabaya. There are 3 ways from Surabaya to Solo. How many possible ways will Santo get from Malang to Solo? (draw the tree diagram!)
Solution Example 4 Albert is caught by a surprise test which consists of 3 true-or-false questions. Illustrate on a tree diagram the possible ways in which he can answer all the questions purely by guess work. Hence deduce the number of possible answers for a 5-question test. (draw the tree diagram for a-3 question test!)
Solution
= … x … = …
Hence there are ….. suits of Ryan’s clothes
= … x … = …
Hence there are … ways from Malang to Solo.
s1
p1
p2
p3
p1
p2
p3
s2
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Using the basic counting principle, we see that there are: … x … x … = … Hence for 5 questions, there are … x … x … x … x … = … = … possible answers.
Example 5 In how many ways can Ami, Beni, and Cinta arrange themselves in a queue?
Solution Let us use 3 boxes to represent the 3 places in queue.
The 1st box can be filled in … ways The 2nd box can be filled in … ways The 3rd box can be filled in … ways By the basic counting principle, the number of arrangements is … x … x … = …
Example 6 There are 5 numbers: 3, 4, 5, 6, and 7. a. Calculate how many different 3 digit odd numbers can be formed from those numbers
without repetition ? b. Calculate how many different odd numbers can be formed from those numbers
without repetition ? Solution
a. For the number to be odd, the last digit must be … , … , or … So there are … ways to fill the last box:
b. (make your box first!)
Example 7 There are 5 numbers: 0, 3, 4, 5, 6, and 7. How many four digits numbers can be formed, that are greater than 2004 but less than 6732? a. If four digits are different? b. If four digits may equal?
Solution a. (make your box first!)
Hence by the basic counting principle, the number of such numbers = … x … x … = …
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b. (make your box first!)
Exercise 1
1. A coin is tossed 3 times. Each toss results in a head or a tail. Find the number of
possible outcomes for the 3 tosses. Illustrate these outcomes on a tree diagram. 2. A quiz on the topic of Statistics consists of 5 multiple-choice questions. Each question
has four choices, of which one is the correct answer. In how many ways can Sania, who is totally unfamiliar with this topic, answer all the questions?
3. Find the number of ways to arrange in a row
a. 5 people b. 10 people
4. Davy has to do the following during her lunch break: take lunch, post a letter, go to the bank, buy the papers. In how many ways can she do all these?
5. Seven boys and five girls are available to form a mixed doubles pair for a tennis match
at Dempo Class Meeting. How many pairs are possible?
6. Three friends decide to have dinner together and then go shopping. Five restaurants are proposed for the dinner and four nearby shopping centres are suggested. How many possibilities are there?
7. Eight people have been short listed for an interview. In how many ways can the
interviewer see them one after another?
8. Calculate how many different numbers that are less than 500 and can be formed from the five digits 3, 4, 5, 6, 7 used without repetition ? and with repetition?
9. How many five-digit numbers can be formed from the digits 2, 3, 5, 7, 8, and 9 if no digit
may be repeated?
10. Calculate how many different 5 digit-numbers can be formed from the ten digits 1, 2, 3, 4, 5 ,6 ,7 ,8 ,9, if: a. every digit can be repeated. b. no digit can be repeated. c. the last digit must be zero and no digit can be repeated.
