Probability and Conditional Probability:
1)A bag contains 15 white balls and 42 gray balls .If one ball is selected at random ,
What is the probability of getting a gray ball.
option:
A) 12/19
B) 19/12
C)13/19
D)11/19
Solution:
General formula for the probability of any event
= Number of successful time of the event / total number of the event
Here number of gray balls =42 ( number of successful event)
Total number of balls in the bag = 15+42=57 ( total number of event)
The probability of getting a gray ball = 42/57=12/19 ( divide by 3 on both Numa rotator and denominator )
Correct ANS:A
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2) A box contain total of 60 pens , of which 25 are green pens and 35 are red pens.
What is the probability of getting a red pen by selecting randomly.
Solution:
Number of red pens =35
Total Number of pens in the box = 60.
The probability of getting ared pen = number of red pens / total number of pens in the box
= 35/60=7/12.
ANS : 7/12.
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3)
| Name | Books | Notebooks |
|---|---|---|
| Peter | 5 | 8 |
| Sam | 10 | 15 |
The above table shows peter and Sam have the number of books and note books .what is the probability that a book is chosen
randomly from Sam.
Option:>
A)3/5
B)2/5
C)5/7
D)2/3
Solution:
Total number of books peter and sams have = 5+10=15
Number of books sam have = 10
The required probability =10/15=2/3
Correct ANS:D
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4)A particular TV show survey result is as follow.
Every day 500 new viewers registered for the show.
What is the probability of new viewers registered for the first three day out of first 20,000 new viewers
registered for the first 10 days.
Option:
A)7/40
B)3/40
C)4/40
D)1/40
Solution ;
Number of new viewers registered for the first 3 days=500* 3= 1500
Total number of viewer registered for the 10 days = 20,000
The probability=1500/20000=15/200=3/40
Corrector ANS:B
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5)A box contains 60 balls.
A pen is chosen at random from the box
The probability of of getting a red pen is 0.35
How may red balls are there in the box.
Solution :
The probability of red ball = .35
Total number of balls = 60
The probability of a red ball = number of red ball / total number of balls
In the box= number of red balls / 60 =0.35
Number of red balls = 0.35 *60= 21
ANS :21 Number of red balls in the box
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6)A candy manufacturing company produce candy of which 15 defective candy for every 320 candy.
If a candy is selected randomly , what is the preobability of getting a good candy.
Options:
A) 61/64
B)51/64C)60/64D)42/64
Solution:
Number of good candies = total number of candy – number of defective candy
= 320 – 15= 305
Total number of candies =320
Probability of getting a good candy = 305/320=61/64
Correct ANS: 61/64
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7) In a educational institution 550 new students joined for the year 2020.
They have to choose either course A, course B or course C.
If the probability of choseing Course B is 0.20 and the probability of choosing course C is 0.32 ,then find out
the number of student chosen Course A.
Solution:
Total number of students in the Institution =550
Probability of students choosing Course b= 0.20
Number of students chosen the course B = 0.20*550=110
Probability of students choosing course C = 0.32
Number of students chosen course C = 0.32*550=176
Number of students chosen course A = 550 110 -176= 264
ANS: 264.
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8) North Carolina Governers from 1800 to 2000
| Age | Number |
|---|---|
| 40-44 | 2 |
| 45-49 | 7 |
| <50-54 | 10 |
| 55-59 | 5 |
| 60-64 | 5 |
| 65-69 | 3 |
The table above gives the number of North Carolina Governers from 1800 to 2000
whose ages at the time they they first took office is with in the interval listed.
Of those Governers who were at least 50 years old when they first took the office,
What fraction where at least 60 years old?
Options:
A) 8/23
B)10/23
C) 9/23
D) 12/23
solution:
The sample space is restricted to Governers who were at least 50 old when they are first took the office
The sum of values in the last four rdws of the table 10+5+5+3 =23 23 is the total number of Governers in the sample spaceThe number of Governers who at lest 60 years old is the sum of the value in the last two rows of the table
That is 5+3 =8
Correct ANS : A
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DATA DISTRIBUTION AND MEASURES OF CENTER AND SPEARD
Arithmetic Mean
The arithmetic mean, commonly known as the average, is calculated by summing all the values in a dataset
and then dividing them by the number of values.
The mean in math and statistics summarizes an entire dataset with a single number representing the data’s
center point . It is also known as the arithmetic mean, and it is the most common measure
of central tendency. Formula for the calculating mean=∑n▒ x k /n
Median
The median is the middle value in a dataset when the values are arranged in ascending or descending order.
If the dataset has an odd number of observations, the median is the middle value.
If the dataset has an even number of observations, the median is the average of the two middle values.
Steps to Find the Median:
Arrange the data in ascending order.
If the number of observation “n” is odd ,Then median =n+1/2 Th observation
If the number of observation ” n” is even then the median is equal to average of the sum of two middle numbers
That is the average of n/2 and(n+1/2)
That is (n/2+(n+1)/2)/2
Mode;
The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal),
more than one mode (bimodal or multimodal), or no mode if all values appear with the same frequency.
Example:
For the dataset [1,2,2,3,4][1, 2, 2, 3, 4][1,2,2,3,4], the mode is 2.
For the dataset[1,1,2222,3333,44,55} ,the modes are 2 and3
For the dataset [1,1,2,2,3,3][1, 1, 2, 2, 3, 3][1,1,2,2,3,3], there is no mode,
as no number repeats more frequently than others.
These measures are essential for understanding the central tendency and distribution of data in various fields,
including statistics, economics, and the social sciences.
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Examples
1) 2,2,2,3,4,4,11
What is the mean. Median and mode of the data set?
Solution:
Mean =2+2+2+3+4+4+11/7 ( sum of the data set divided by number of
number of the data set}
Mean= 28/7=4
Median ( middle observation)
Arrange in assenting order 2, 2, 2,3 ,4,4 11
There are 7 observation in the data set
,p> Meadian =7+1/2 =8/2 4 (odd number observation4 Th observation of the data set is the median of the data set
Median = 3
Mode: (most occurring observation)
In the above data set “2” is occurring 3 times
There fore 2 is the mode of the observation.
Mean = 4
median= 3
mode=2———————————————————————-
2)Data set X 5, 6, 7, 8,10
Dataset: 6, 8, 9,10, 12
Which of the following statement is correct for the Data set X and Y.
Options:
A)Mean of each data set is 7.2
B) mean of each data set is 9
C) Mean of data set X is greater than mean of data set Y
D)Mean of data set X is lesser than mean of data set Y
Solution:
Mean of the data set X +(5+6+7+8+10)/5 = 36/5 =7.2
Mean of data set Y =(6+8+9+10+12)/5 =45/5 =9
Correct ANS: D
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3) Data set A consists of a height of 40 trees has a mean of 30 feet
Data sat consists of a height of 30 trees has a mean heigh of 31 feet
Data set consists of consists of height of 70 trees of set A and B.What is the mean of Data set C.
Solution:
Total height of 40 treesof the data set with 30 feet mean = 40 × 30 = 1200 feet
Total height of30 trees of the data set B with mean of 31 feet =30×31=930feet
Mean of data set C= (1200+930)/(40+30) =2100/70 =30.4
ANS: 30,4 feet
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Solution:
General form of Any consecutive 11 integers can be written as K, K+1, k+2, k+3, k+4, k+5, k+6, k+7, k+8, k+9, k+10
.Mean of this integers is +(k+k+1+k+2+k+3+k+4+k+5+k+6++8+k+9+k+10)/11 = (11k+55)/11
Mean= k+5
Median = the middle number of the 11 integers
= (11+1)/2 = 12/2 =6 Th inter ( it is an odd number of integers)
=K+5
Therefore 3a+2b = 3(k+5) +2(k+5) =3k+15+2k+10 = 5k+25
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5) Ages of 38 antedates attended an HR interview in a company
is given in the table
| Age | Frequency |
| 35 | 12 |
| 36 | 10 |
| 37 | 5 |
| 38 | 4 |
| 38 | 4 |
| 39 | 2 |
| 40 | 2 |
| 41 | 3 |
The above table shows distribution of ages of 38 antedates who attended a HR interview
which of the following gives the correct order of the Mean, median, mode of the ages
options;
A) mode < median < mean
B) mean < median < mode
C)median< mode < mean
D) mode>median>mean
Solution:
Mean of the above data = sum of the data values is divided by the number of the data values
(35×12) +(36×10) +(37×5) +(38×4) +(39×2) +(40×2) +(41×3)] ÷ [12+10+5+4+2+2+3]
= [ 420+360+185+142+78+80+123] ÷ [38]
=1988 ÷38 =52.3
Mean =52.3
Median:
Is the middle number of the data set
38 is even , so it has two middle+1 number.
That is 38/2 and 38/2
Median = average value of the 19th and 20 th value
in the above table 19 th and 20 th value = 36 and 36
Median is 36
Mode: the most occurring frequency
Here the most occurring frequency is 35.
Mean=52.3, median = 36 and mode=35
Correct ANS: A
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6) the given table summarizes number of books sripooirisuju has
| Number of pages | frequency |
| 60 | 3 |
| 61 | 5 |
| 62 | 3 |
| 63 | 5 |
| 64 | 6 |
| 65 | 7 |
| 66 | 8 |
| 67 | 1 |
the above table gives number of pages in each book .
If they buy a new book have 70 pages in it
How it will affect the mean and median their book collection?
Options:
A) Mean and median both will decreases
B) median will increase and mean will decrease
C) mean will increase and median remain the same
D) mean and median will both decrease
solution:
it is in general the added new book has more pages
the will increasethe median is the middle number of the above data set. the data set has 38 books, newly added another book total number of books they have =38+1=39
39 is odd number, so the median = (39+1)/2=40/2=20 th book
from the frequency table the 20 th book has 50 pages
the new median id 50
the median of the frequency table before the new addition
the number of books is 38 , so the middle number is 19 and 20 th number
19 th and 20 th book is 50 and 50
there ius no change in median
but there is a increse in the mean
Correct ANS:C
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7) table of symmetric distribution
| Valuer | Frequency |
| 1 | 2x |
| 2 | 4x |
| 3 | 6x |
| 4 | 4x |
| 5 | 2x |
The frequency distribution above summarizes a set of Data
where x is a positive integer Discus the mean and median of this data set
option:
A) mean is greater than median
B) mean and are the same
C) median is greater than mean
solution
From the above frequency table we can under stand that the middle value of the set is 6x
we can also understand that above the middle valu and below the meddle value are the same it means that this is a symmetric distribution
in any symmetric distribution the mean and median are same ( by definition of symmetric distribution)
The ANS:B
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Percentages:
8) what is the 45% of 500?
Options: A)250 B) 450 c) 225 D) 150
Solution;
45% means for 100 it is 45
For 500 = 45/100×500=45 ×5= 225
Correct ANS: C
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9)Which of the following represent the resuit of a number increased by 42%
Option: A) .43 of the number B) 1.42 of the number C) .57 of the number D) None of these.
Solution:
Here the number is uncreased by 42%the number
This can be written by (1+42/(100))) ×the number.
That is = (1+0.42) = 1.42 of the number
Correct ANS :B
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10) The cost of a TV is $56o with the tax(taxis included)
Sales tax for the TV is 12 %. What is the cost of TV with out TAX.
Solution:
Sales price Of TV = TV cost + sales Tax
Let tv cost = $x
Sales tax = 12%
Sales Tax for $ x = 12/100×x=0.12x
Therefore TV sales price = TVcost + sales Tax= x +0.12x=1.12x
560 =1.12 x(Divide both side by 1.12)
1.12x/1.12=560/1.12
x =$ 500
ANS:$500
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11)The number a is 50% greater then the positive number b.The number
is 40 % leaser than the a .the number c is how many times b?
Solution:
The number a is 50% greater than b
Therefore a=(1+50/100) b =(1+0.50)b=1.5b
It is also given that numbef cis 40% lesser then the number a
Thai is c=(1-40/100) a =( 1-0.40) a =0.60a
C= 0.6a,but a =1.5 b
Therefore c = 0.6×1.5b=0.9b
ANS: c is 0.9 times of b
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________________________________________—Histogram of Students per Subject
Ratio and Percentages
14)A number p increased by 12%. if p=60.what is the increased number?
Option:
A)107.2
B)106
C)110
D)102
solution:
Solution:
Here it is given p=60
P increased by 12%= 12/100
12/100 of 60 =12/100 ×60=720/100=7.2
The increased number=100+7.2=107.2.
Corrcet ANS:A
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15)15 is a% of 50 ,what is the value of p?
Option
A)35
B)40
C)30
D)25
Solution:
15 is a% of 50 can be written as
15/50=a/100
That is a/100 =15/50
15/50=a/100
That is a/100 =15/50
Multiply by 100 on both side
Therefore a/100×100=15/50×100
So a=15×100/50=30
Multiply by 100 on both side
Therefore a/100×100=15/50×100
So a=15×100/50=30
Correct ANS:C
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16) A TV is sold for $ 918with 2% tax .What is the price Before Tax.?
Option:
A)900
B)950
C)800
D)850
solution:
Solution: Let the cost of TV before tax is $ x.
Tax =2%
Sale price =x +2% Tax
=x+0.02x
=1.02x
It is given sale price =918
Therefore 1.02 x = 918
X=918/1.02=$900
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17) An Organization form a committee to discusses about their future plan of the organization.
In that committee consists of35% are worker ,20% are officers ,25% shareholder and 10 distributors,
Find the number of Shareholders.?
Option:
A)50
B)55
C)60
D)40
Solution:
Number of Distributors =10
The percentage of distributor= 100 –35-20-25=20
Let the number of people in the committee=p
Therefore 20% of x= 20/100×p=0.2p
By equating the number of distributor10 to % of Distributors
0.2p= 10 ( divide both side by 0.2)
0.2p/0.2=10/0.2
P=10/0.2 (Both denominator and numerator Multiply by10)
P= 10×10/0.2×10=100/2=50
The committee consists of 50 people.
Correct ANS:A
___________________________________________________-18)The ratio of 10 t015 is equal to x to 3 and
The ratio of 12 to y is equal to 10 to 15.
Find the ratio of x to y.
Option:
A)1 to 3
B)3 to 1
C)3 to 9
D)9 to 3
Solution:
The ratio of 10 to 15 is equal to x to 3
Can be written as 10/15=x/3
Multiply by 3 on both side
x/3×3= 10/15×3
×x=10×3/15=2
The ratio 12 to y is equal to 6 to 4
Can be written 1y2/y=6/4 Multiply by y both sides
12/y×y=6/4×y
12= 6y/4
6y/4=12 multiply by 4 on both sides
6y/4×4=12×4
6y= 12×4
Y= 12 ×4/6 (multiply by 6)
Y=8
The ratio x to y is equal to 2 to 6
The ratio x to y is equal to 1 to 3
Correct ANS:A
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19) A document writer can type 32 words per minutes
What is rate per hour?
Option:
A)900words
B)950 words
C)1000 words
d)680 words
Solution:
Typing rate per minutes= 15words
Typing rate per hour= 15×60= 900 words
Typing rate per minutes= 15words
Typing rate per hour= 15×60= 900 words
Correct ANS:A
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Question number19
| x | y |
|---|---|
| 5 | 6 |
| 10 | p |
| 15 | 18 |
| 20 | 24 |
| 25 | 30 |
In the above table ,thge ratio of x to y is constant for all x and coordinates
What is the value of p in the above table?
Option:
A)12
B)15
C)12
D)10
Solution:
in the above table the ratio of x to y is 5 to 6
Thais x/y=5/6——(1)
consider the ratio of x to y is 10 to p
that is x/y=10/k——–(2)
compare (1) and (2)
we can have 10/k=5/6
Multiply by k on both side
we got 10k×/k=5k×/6
That is 10=5k/6
Multiply by 6 on both side
10×6=5k6/6
60=5k
5k=60, divde by 5 on both side
5k/5=60/5
k=12
Correct ANS:A
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