Mathematics for Junior High School

By Marsigit

C. Probability with relative Frequency

Probability can be explained from the definition of relative frequency. See the explanation below!
• In the experiment of tossing a coin, the ratio between the occurrence of the head and the total number of the experiment is called the relative frequency of obtaining the head.
• If in the experiment, the coin balances enough then the more the experiment is taken out, the more the relative frequency of getting a head is close to .
• In this case, we can tell that in the experiment of throwing a balance coin, the probability of obtaining a head (P(G)) is .
If n is large enough, then the probability of event A is
Where f represents the frequency of event A and n represents the total number of experiment.

Example:
From the experiment of 40 times tossing of 1 coin , the occurrence of head is 22 times. Find the relative frequency of obtaining a head !
Solution :
The total number of experiment (n) is 40 times.
The frequency of the occurrences of the head, f=22
So, the relative frequency of the occurrences of the head is=



Exercises
1. In the experiment of rolling once a six-sided dice , find :
a. the probability of occurrences of the 3-spot side
b. the probability of occurrences of the odd-spot side
2. From tossing a coin 30 times, there are 16 times tail occurrences. Find :
a. the relative frequency of obtaining of the tail
b. the relative frequency of obtaining of the head

3. Expectation Frequency
In an experiment, if A is an event and the probability of event A is P(A) then the expectation frequency of event A for n times experiments is determined by the formula stated below


Example :
A coin was tossed 30 times (assume the coin balances ). What is the expectation frequency of obtaining a head ?
Solution :
The total number of tossing is 30 times. If H is an event of obtaining a head, the probability of obtaining a head is . Hence the frequency of expectation of obtaining a head is

So, the expectation frequency of obtaining the head is 15 times.

Exercises

1. Two coins were tossed together 10 times. What is the expectation frequency of obtaining a head from the first coin and the tail from the other coin?
2. A six-sided dice was thrown 30 times. What is the frequency of expectation of occurrences of the even-spot side?
3. A coin and a dice were rolled together 4 times. What is the expectation frequency of obtaining a tail from the coin and even-spot side from the dice?
4. Two dice were rolled together. How many experiments that have been taken out if the expectation frequency of occurrences of the same side from both of the dice is 10 times?



4. Compound Events

Compound event is an event that contains 2 or more events. In the next explanation, we will discuss about the probability of some compound events contained 2 or more events.

A. Probability Of Non Mutually Exclusive Event

Event A and event B are non mutually exclusive if . Consider the Venn diagram of event A and B that are non mutually exclusive events below
….
The probability of 2 events ( event A and event B) that are non mutually exclusive is as follows


Remember:
is an intersection operation in sets algebra
is an empty sets that is a set which has no elements.
is an union operation in sets algebra
Note:
In mathematics, the union operation is presented by term 'or'. While the intersect operation is presented by term 'and'. For example, P(AUB) is stated as the probability of event A or event B. While P(A∩B) is stated as the probability of event A and event B..

Example
A six-sided dice is rolled once. Find the probability of occurrences of the primary number spot side or the sides with spots less than 5.
Solution:
The sample space of the results of rolling a six sided dice once is S={1,2,3,4,5,6}. So, the total member of sample space S is n(S)=6
If A is an event of obtaining primary number spot side, then A={2,33,5}. So, the total member of event A is n(A)=3.
If B is an event of obtaining the sides with spot less than 5. So, the total member of event B is n(B)=4.
Hence, we can conclude that


So, the total member of event is . So, A and B are non mutually exclusive events.
As a results

The probability of obtaining primary number spot side or sides with spot less that 5 is 5/6

B. Probability of Mutually Exclusive Events
Event A and event B is called mutually exclusive event if . See the Venn diagram of event A and B that are mutually exclusive event below
….


Example

A six sided dice is thrown once. Find the probability of occurrences of the sides with spots less than 3 or the sides with spots greater than or equal to 5!

Solution :
The sample space of the results of throwing a six sided dice once is S={1,2,3,4,5,6}. So the total member of sample space S is n(S)=6
If A is an event of occurrences of the side wit spots less than3, then A={1,2}. The total member of event A is n(A)=2
If B is an event of occurrences of the side with spots greater than or equal with 5 then B={5,6}. The total member of event B is n(B)=2
We can conclude that . So A and B are mutually exclusive events.


as a results

So the probability of occurrences of sides with spots less than 3 or the sides with spots greater then or equal with 5 is 4/6

c. The Probability of Independent Events

Event A and event B are called independent events if the occurrence or non-occurrence of event A is not in any way influenced by the occurrence or non-occurrence of another event. The probability of independent events is as follows


Examples
A basket contains 4 oranges and 6 apples. The other basket contains 5 oranges and 15 apples. What is the probability of obtaining an orange from the first basket and an orange too from the other basket?
Answer :
If,
S1 is sample space of fruits in the first basket, then the number of fruits in the first basket is n(S1)=10
S2 is a sample space of fruits in the second basket, then the number of fruits in the second basket is n(S2)=20
A is an event of obtaining an orange from the first basket n(A)=4
B is an event of obtaining an orange from the second basket, then n(B)=5

So the probability of obtaining an orange from the first basket and an orange from the second basket is 1/10

Exercises
1. A six sided dice was rolled once. Find:
a. the probability of obtaining the even-spot side or the odd-spot side
b. the probability of obtaining the even spot side or the sides with spot less than 3
2. From 52 cards, a card is taken randomly. Determine:
a. the probability that the red card or black card is selected
b. the probability that the red ace card or black ace card is selected
c. the probability that the black card or the 9th card is selected
d. the probability that the red card or the black ace card is selected
3. There are 80 students from a certain school joining some extracurricular programs. It was found that 25 students were joining karate, 15 students were joining swimming and 10 students were joining both karate and swimming program. While the rests were joining the other programs. If we select randomly 1 student from the students that were joining the extracurricular programs, what is the probability that the students which were joining karate or swimming program is selected?
4. Class IX A consists of 40 students. It is found that 10 from 40 students like Biology subject, 20 like English and 5 students like both Biology and English. The other students like the other subject. If we select randomly 1 student from class IX A, what is the probability that the students that like Biology or English is selected?
5. Two six-sided dice were rolled once together. Determine:
a. the probability of occurrences of 1 spot side of the first dice and the primary number spot side of the second dice.
b. the probability of occurrences of 2 spot side of the first dice and 3 spot side of the other dice.






Exercise Chapter 3
A. Chose the correct answer

1. Mrs. Tuti tastes 1 spoon of soup from a bowl of soup. The population from the illustration above is
a. 1 bowl of soup
b. 1 spoon of soup tasted by mrs Tuti
c. 1 pan of soup
d. 1 plate of soup
2. Given the data presented as below
….
The average of the data is....
a. 18
b. 18.25
c. 18.50
d. 18.75

3. Given the data of the student’s age from a certain organization is presented below
The median of the data is ....
a. 13
b. 13.50
c. 15.50
d. 16
4. The data of the weight of a group of athletes is presented below...
The modus of data above is ....
a. 42
b. 44.5
c. 47
d. 42 and 47
5. The average of the art test score from a certain student group consist of 5 students is 75. After a new member join the group, the average turns into 73. The art test score of the new member is....
a. 73
b. 70
c. 63
d. 60

6. See the data of English test score from 50 students presented in the bar diagram below.
The modus of the data of the English test score is ....
a. 5
b. 7
c. 9
d. 5 and 7
7. The money ( in rupiahs ) owned by a student in a week is presented in the diagram below.
The average of the student's money in a week is ....
a. Rp 6000,00
b. Rp 6200,00
c. Rp 6500,00
d. Rp 6800,00
8. If 2 coins were tossed once then the probability of obtaining a tail from one of the coins is ...
a. 1/4
b. 1/2
c. 3/4
d. 1

9. Tono checks his 2 game CD that has not being used for a long time. He wants to know whether his CDs are still working or not. The number of possible outcome that might occur from the checking is....
a. 1
b. 2
c. 4
d. 6

10. The total member of a certain Arisan group is 40 people. Every round, there are 4 people that get the Arisan money. The probability of a member getting the money in the first round is ....
a. 1/10
b. 1/10
c. 1/4
d. 1

11. In the experiment of throwing a coin three times, B is an event of obtaining a tail once. The B event can be expressed as….
a. {T}
b. {THH}
c. {THH, HTH, HHT}
d. {THH, HTH, HHT, HHH}
12. If from 1 pack of cards is taken 1 card randomly, then the probability of obtaining an Ace is ….
a.
b.
c.
d.
13. A box contains 4 red marble, 5 white and 6 green ones. If 1 marble is taken randomly, the probability of occurrences of obtaining a red marble is ….
a.
b.
c.
d.
14. A manager from a certain company gets information that 3 of 100 products from that company are damaged. If 1 product from that company is taken randomly, the probability that the product is good is ….
a.
b.
c.
d.
15. A coin and a six-sided dice are rolled together. The probability of obtaining the 5 spot side is ….
a.
b.
c.
d.
16. Two dices are thrown together. The probability that both of the sides are same is ….
a.
b.
c.
d.
17. A six-sided dice is rolled 18 times. The relative frequency ( the expectation frequency) of obtaining a side with spots less that 4 is ….
a. 2
b. 3
c. 6
d. 9
19. A six sided dice and a coin were rolled together once. The probability of occurrences of obtaining a primary number spot side and the tail from the coin is ….
a. 0.25
b. 0.5
c. 0.75
d. 0.85
20. From 52 cards we took 1 card randomly. The probability of getting a black Ace or a red card is ….
a.
b.
c.
d.
B. Do the exercises below
1. The data of favorite sports of 120 students is presented as a pie diagram below.
a. How many students that like badminton ?
b. What is the modus of student favourite sports? Explain your answer
2. The try out score of Science National Examination from a certain school is presented in table below
a. How many students whose score is more than 5?
b. Calculate the average of the try out score !
c. Determine the median and modus of the data above ?
3. A student is observing whether 3 laboratory equipments are still working or not.
a. Determine the sample space of the observation
b. If A is an event of getting 2 equipment damaged, write the member of event A !
c. Determine the probability of event that 1 equipment is damaged !
4. Given 2 box, each contains 5 balls. The balls in every box were labeled 1 to 5. Then, from each box was taken 1 ball randomly all at once.
a. What is the probability of obtaining that the balls of each box have the same label.
b. What is the probability of obtaining an odd numbered ball from the first box?
5. The probability of a student to be chosen as representatives in the scientific paper competition for teenagers in a certain school is 0.025. How many students that will be the representatives in the competition if the total number of students joining is 720?

Evaluation 1-3
A. Chose the correct answers
1. Two triangles below are congruent in the case of the relation….
a. side, side, side
b. side, side, angle
c. side, angle, angle
d. angle, angle, angle
….
2. Two triangles below are congruent in the case of the relation ….
a. side, side, side
b. side, side, angle
c. side, angle, angle
d. angle, angle, angle
….

3. Two triangles below are congruent in the case of the relation….
a. side, side, side
b. side, side, angle
c. side, angle, angle
d. angle, angle, angle
….

4. Given ABC and CBD. If AB=10 cm and BC=7 cm then the length of BD is ....
a. 10 cm
b. cm
c. cm
d. 4.9 cm
….

5. Given ABC and PQR are similar. CAB=RPQ, ABC= PQR, and BCA=QRP then the length of PR is ….
a. 7.14 cm
b. 3.5 cm
c. 0.56 cm
d. 0.14 cm
….
6. Given ABC and CBD are similar. CAB=DCB=90o and CBA=DBC. If AC=6 cm and BC=10 cm then the length of AB and the length of CD are ….
a. AB=7.5 cm and CD=8 cm
b. AB=8 cm and CD=4.8 cm
c. AB=8 cm and CD=7.5 cm
d. AB=8 cm and CD=13.3 cm
….

7. Given AOB and XOY are similar. If AO=2 cm, XA=3cm, AB=4 cm, and BY=5.1 cm then the length of XY and the length of OB are ….
a. XY=3.4 cm and OB=1.6 cm
b. XY=3.4 cm and OB=10 cm
c. XY=1.6 cm and OB=3.4 cm
d. XY=10 cm and OB=3.4 cm
….
8. Given ABC and DEC are similar. If CD=5 cm and AD=3 cm then the value of is ….
a.
b.
c.
d.
….

9. Given ABX and CDX are similar. If AX= 8 cm XC=10 cm, and BD=27 cm then the length of DX is ….
a. 12 cm
b. 13 cm
c. 14 cm
d. 15 cm
….

10. Given ABC and DEC are similar. If BCA:CAB:ABC = 1:2:1 then the value of DEC is ….
a. 25o
b. 30o
c. 45o
d. 50o
….

11. A cylinder has a radius 6 cm and the side area is 37.86 cm2. The height of the cylinder is ....
a. 12 cm
b. 10 cm
c. 2 cm
d. 1 cm

12. Oil 4.71 liter was poured into a can formed cylinder so the cylinder is filled with the oil until 15 cm high. The diameter of the cylinder is ....
a. 10 cm
b. 15 cm
c. 20 cm
d. 30 cm

13. The side area of a cylinder is 440 cm2. If the height of the cylinder is 10 cm then the top and bottom area of the cylinder are ....
a. 308 cm2
b. 154 cm2
c. 88 cm2
d. 44 cm2

14. A biscuit can has diameter 20 cm and height 10 cm. The surface area of the biscuit can is ....
a. 12,560 cm2
b. 1,256 cm2
c. 628 cm2
d. 125.6 cm2

15. A trumpet made from cardboard and formed a cone has a diameter of 14 cm. If the slant height is 30 cm then the area of the cardboard needed to make the trumpet is ....
a. 1,320 cm2
b. 660 cm2
c. 132 cm2
d. 66 cm2

16. A cone has a slant height of 7 cm and radius 3.5 cm. The surface area of the cone is ....
a. 1,155 cm2
b. 770 cm2
c. 115.5 cm2
d. 77 cm2

17. If the volume of a cone having perpendicular height 9 cm is 462 cm3 then the base area of the cone is ....
a. 44 cm2
b. 154 cm2
c. 1,386 cm2
d. 7,546 cm2

18. Mom has a mold to make jelly formed a half sphere and having a diameter of 21 cm. The volume of the jelly if it was made using the mold is ....
a. 38,808 cm3
b. 19,404 cm3
c. 4,851 cm3
d. 2,425.5 cm3

19. A ball is made from woven rattan. If the total area of woven rattan used to make the sphere is 616 cm2 then the diameter of the ball is ....
a. 4.9 cm
b. 7 cm
c. 9.8 cm
d. 14 cm

20. An aquarium formed a sphere has diameter of 35 cm. The volume of the aquarium is ....
a. 67,375 liter
b. 22,458.33 liter
c. 67.375 liter
d. 22.458 liter
Do number 21-22 based on the table below. We were given a data of weight of an apple from some different trees below.
….

21. The total number of apples that are being measured is….
a. 43 apples
b. 44 apples
c. 48 apples
d. 50 apples
22. If the apples that have weight less than 80 gram are not sold then the total number of apples that are sold is ….
a. 33 apples
b. 26 apples
c. 24 apples
d. 17 apples

Do number 23-25 based on the circle graph below. The data of student favorite activities in their spare time is presented below.
….
23. The value of x is …..
a. 100o
b. 110o
c. 120o
d. 130o

24. The percentage of students that like reading is ….
a. 80 %
b. 40 %
c. 33.33%
d. 22.22%

25. If the number of students who like reading is 40 people, then the number of students who like watching TV is ….
a. 50 students
b. 60 students
c. 80 students
d. 120 students

26. We were given a data of the total number of goal occurrences of 15 soccer games in a certain season
….
The median of the data above is ….
a. 1
b. 2
c. 3
d. 4

27. A box contains 4 red marble, 3 white, and 5 green ones. If we took 1 marble randomly then the probability of obtaining a white marble is ….
a. 0.25
b. 0.33
c. 0.42
d. 0.60

28. If the probability of a student getting scholarship is 0.125 then the probability of a student not getting that scholarship is …
a. 0.875
b. 0.785
c.
d.

29. A six sided dice and a coin were rolled together. The probability of occurrences of odd-spot side of the dice and a tail of the coin is ….
a. 0.75
b. 0.5
c. 0.25
d. 0.125

30. A card is taken randomly from 52 cards. The probability of getting an Ace card is ….
a.
b.
c.
d.
B. Solve the following problems

1. In the afternoon, a boy who has height 1.5 m has a shadow with length of 4.5 m. Tina’s height is 1.3 m. What is the length of Tina’s shadow if she stands in the same time and the same place as the boy?
2. See the picture below. A fisherman wants to know about the distance of his friend boat that sailing in the ocean from the coast line. Based on the position of the boat, the fisherman is located in point D. Then he make another points ABC along the coast according to the sketch below. Determine the length of AA'!
….
3. Prove that ABC and ADC are congruent!
4. A cylinder has maximum capacity of 2.156 liter. If the diameter of the cylinder is 14 cm, then calculate the height of the cylinder!
5. A traditional cap formed a cone has a slant height of 28 cm and radius 21 cm. It is made from woven rattan. Find the area of the woven rattan used to make the traditional cap!
6. A bowl is formed half-sphere with diameter 14 cm. Calculate the volume of the bowl!
7. A ball has a circumference of the mid circle of 66 cm. Determine the total area of material used to cover the leather!
8. A student has a collection of 50 books, consists of 20 school text books, 10 fiction books, 15 motivation books, and 5 biography books of world famous person. Draw a circle graph to illustrate the information!
9. A trader mixes 10 kg oranges that priced Rp 15.000,00 per kg and 5 kg oranges priced Rp 10.000,00. What is the average price of that mixed oranges now?
10. In a certain school, there are 100 students who propose for scholarship. Each student has the probability of 0.25 to get the scholarship. How many students who will get the schoolarship?