Probability Maths

Introduction to probability maths:

Probability is a word that is been used commonly in our day today life without going into the details of its actual meaning. We come across statements like:

(i) Probably it may rain to day.

(ii) He may possibly join politics.

(iii) Argentina team has good chance of winning world cup.

(iv) She is probably right.

In these kinds of statements, we generally use the terms: possible, probable, chance, likely etc. All these terms convey the same meaning that the event is not certain to take place or, or other words there is uncertainty about the occurrence of the event in question.

Now let us see few problems of this kind..


Example problems on probability maths:


1. Find the probability that a leap year selected at random will contain 53 Sundays.

Soln: We know that in a leap year there are 366 days.

They can be written as, 366 days = 52 weeks and 2 days.

Thus a leap year has always 52 Sundays.

The remaining two days cane as follows:

(i) Sunday and Monday

(ii) Monday and Tuesday

(iii) Tuesday and Wednesday

(iv) Wednesday and Thursday

(v) Thursday and Friday

(vi) Friday and Saturday

(vii) Saturday and Sunday.

Clearly, there are seven events are associated with this random experiment. Let X be the event that a leap year has 53 Sundays. This is possible if the last two days are either Sunday and Monday or Saturday and Sunday.

Therefore the required probability is `2/7` .

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More example problems on probability maths


2. Let X and Y throw a pair of dice. If X throws 9, find Y’s chance of throwing a higher number.

Soln: To throw a value greater than 9, we have the following possibilities:

{(4,6),(5,5),(5,6),(6,4),(6,6),(6,6)}. Therefore, the number of chances is 6.

When two dice are thrown, the total possibility is 36.

Hence the required probability is `6/36 = 1/6` .

3. A letter is chosen at random from the letters of the word “ASSASSINATION”. Find the probability that the letter chosen is a consonant.

Soln: The word “ASSASSINATION” has 13 words.

Here the consonants are {s, s, s, s, n, t, n}. Hence we have 7 consonants.

Therefore the probability to get a consonant is `7/13` .

Decimal Skill Practice

Introduction of decimal skill practice:
Decimal is one of the types of the numbers system. The point is called decimals. Here we are going to discuss about adding decimals, subtracting decimals, multiplication decimals and dividing decimals. And we are going to learn how to improve our decimal skills and practice problems here .Decimals are also fractional numbers. For example 0.5 is the same as the fraction 5/10.


Decimal skill practice:


There are 4 different types of decimal skills.

That is ,

Adding and subtracting decimal skills.
Multiplication decimal skills.
Division decimal skills.


Addition and subtracting decimal skill practice:

Its commonly used to find the total of two numbers. Generally addition and subtraction of decimals is totally based on adding and subtracting whole numbers. But here we need to concentrate to place the decimals

Example of adding decimals:

Example 1:

Add the decimal value: 12.74, 15.11

Solution:

4 2 . 15

1 3.  1 1    (+)

-------------

5 5  .  2 6



Example of subtracting decimals:

Example 2:

Subtract the decimal value 243.87, 23.56.

Solution:

3 2 1 . 8 7

1 5 6 . 5 6   (-)

------------------

1 6  5 . 3  1


Example problem for multiplication and division decimal skill practice :


Example of multiplication decimals:

Multiplication is a product of two terms.

Example 1:

Multiply 7.9 * 1.3

Step 1: First remove the decimal point.

Step 2: And then perform multiplication operation.

Step 3: lastly add the decimal point in the answer.

Solution:

79

13   (*)

----------

2 3 7

7 9

-----------

1 0 2 7

Answer: Form the given question we have two decimal points. So we need to put the 2 decimal points to the answer So, the answer is 10.27.

Dividing decimals:

Its also the equal of whole number division.

Example 2:

10.5 divided by 0.5

Step1: Convert the decimal values in to the whole number.

Step 2: So, multiply 10 with both top and bottom.

Step 3: `(10.5 times 10)/(0.5 times 10)` .So we get `(105)/(5)`

Step 4: Here we can perform the division process.

Step 5: Therefore the answer is 21.

Practice Integers

Introduction to practice integers:

In mathematics, integers is one interesting topics in number representation. Integer has a set of numbers in which there are two types of integers that are non negative integers and negative integers. Integers have complete entity or unit.Integers perform different types of arithmetic operations such as addition, subtraction, multiplication and division. Let us see some example problems and practice problems.

Example integers:

7254, - 8564, 0 etc,


practice integers - Example problems:


Different types of example problems for integers are,

Example Problem for addition:

Perform addition for the given two integers

1087 + 5892

Solution:

Given two integer are

1087+ 5892

Here we add 1087 into 5892, and then we get the result

=1087 + 5892

=6979

Solution to the given two integers is 6979.

Example Problem for subtraction:

Perform arithmetic operation subtraction for the given two integers

854 and 789

Solution:

Given two integer numbers are

854 – 789

Here we subtract 854 into 789, and then we get the result

=854 – 789

=65

Solution to the given two integers is 65.

Example Problem for multiplication:

Perform arithmetic operation multiplication for the given two integers

264 × 147

Solution:

Given two integer numbers are

264 × 147

Here we multiply 246 into 147, and then we get the result

=264 × 147

=38808

Solution to the given two integers is 38808.

Example Problem for division:

Perform arithmetic operation division for the given two integer numbers

735/15

Solution:

Given two integer numbers are

735 / 15

Here we divide 735 by 15, and then we get the result

=735/ 15

=49

Solution to the given two integers is 49.

Example Problem for addition:

Perform arithmetic operation addition for the given two integer numbers

- 364 + 785

Solution:

Given two integer numbers are

- 364 + 785

- 364 is negative number and 785 is positive number

Here we add - 364 into 785, and then we get the result

= - 364 + 785

=461

Solution to the given two integers is 461.


practice integers - Practice problems:


Perform arithmetic operation for the given two integers are,

i). 786 – 225

ii). 621 + 106

iii). 45 × 21

iv). 1250 / 50

v). – 897 × 154

vi) – 456 + 264

Solution:

i). 561

ii). 727

iii). 945

iv). 25

v) – 138138

vi) – 192

Mixed Numbers Practice

Introduction for Mixed numbers:

In a fraction if numerator is greater than denominator then this kind of fraction is know improper fraction the improper fraction in standard form is known as mixed number.

A mixed number consists of

A whole number.

Proper fraction.

A `b/c`

Here A is whole number.

Now let see problems on mixed numbers operations.


Mixed numbers practice problems:


Practice problem 1.

Find the sum of two mixed number 6`1/3` and 7`1/3`

Solution:

The given mixed numbers are  6`1/3` and  7`1/3`

Initially to perform any operations on mixed numbers we must convert it to fraction

6`1/3` in fraction

Multiply 6 and 3 and add with 1

6`1/3` = `(19+1)/3`

=`20/3`

Now Convert the 7`1/3`

Multiply 7 and 3 and add with 1

7`1/3` =` (21+1)/3`

=`22/3`

6`1/3` +7`1/3` =`20/3` +`22/3`

=`(20+22)/3`

= `42/3`

This can be simplified has 14

Practice problem 2.

Find the difference of two mixed number 8`1/3` and 9`1/3`

Solution:

The given mixed numbers are 8`1/3` and 9`1/3`

Initially to perform any operations on mixed numbers we must convert it to fraction

8`1/3 ` in fraction

Multiply 8 and 3 and add with 1

8`1/3`  =` (24+1)/3`

=`25/3`

Now convert  9`1/3` in fraction

Multiply 9 and 3 and add with 1

9`1/3` = `(27+1)/3`

= `28/3`

8`1/3` - 9`1/3` = `25/3` -`28/3`

=`(25-28)/3`

=`-3/3`

This can be simplified has  -1

Practice problem 3.

Find the product of 5` 1/3` and 6`1/3`

Solution:

The given mixed numbers are 5`1/3` and 6`1/3`

Initially to perform any operations on mixed numbers we must convert it to fraction

5`1/3` in fraction

Multiply 5 and 3 and add with 1

5`1/3` =`(15+1)/3`

=`16/2`

Now convert the nextmixed numbers  6`1/3 `

Multiply 6 and 3 and add with 1

6`1/3`  =` (18+1)/3`

= `19/3`

5`1/3` `xx` 6`1/3` =`16/3` `xx` `19/3`

= `(16xx19)/(3xx3)`

= `304/9`

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Some more mixed numbers practice problems:

Practice problem 4.

Find the sum of two mixed number 5`1/5` and 6`1/5`

Solution:

The given mixed numbers are 5`1/5` and 6`1/5`

Initially to perform any operations on mixed numbers we must convert it to fraction

5`1/5` in fraction

Multiply 5 and 5 and add with 1

5`1/5` = `(25+1)/4 ` =`26/4`

6`1/5`  `rArr` Multiply 6 and 5 and add with 1

6`1/5`  =` (30+1)/5`

= 3`1/5`

5`1/5` + 6`1/5` =`26/5` +3`1/5`

=`(26+31)/5`

= `57/5`

This can be simplified has 11.4

These are some examples of mixed numbers.

Multiplying Rational Expressions

Multiplying rational expressions I (math)

A rational expression  is an algebraic expression  which is in the  form P/Q, where P and Q are simpler expressions P and Q are usually polynomials.  The denominator Q is not zero. It is the quotient of two polynomials. It is important to remember that the divisor cannot be zero. A quotient is the answer to a division problem. Polynomials are expressions which has the sum of powers with at least one variable.  This variable is multiplied by coefficients.

Examples of rational expressions are :

2/5 ,    3x + 9    ,        x^2 + 6x + 3
X + 3               x^2 +x +2

Multiplication Rational Expressions

When  we multiply the rational-expressions the numerators  and the denominators  are to be multiplied.   If P,Q,R  and S polynomials   then

P / Q   .  R / S  =   PR / QS  When Q is not equal to zero and S is not equal to 0.

Here we multiply the  numerators and the denominators.   Before multiplying it has to be seen if they can be reduced.   If reduction is possible then  it has to be done before multiplication.

Multiplying Rational Expressions Examples

Multiply the rational expression given below

18a3       x     5b4

20b2                   6a

Before multiplying the above expression it can be simplified.  Here we can cancel the numerator with the denominator or denominator with the numerator


18 a3       x     5 b4

20 2                   6a

18 can be divided by 6 and we get 3 and   a3 divided by a  = a2   and  b4 /b2 =b2 .  Thus we have 3a2 b2 in the numerator and 4 in the denominator.



3a2        x   b2                                           3a2  b2
=                 4
4               1

Multiply 3a2 with b2 and 4 with 1.  We get 3a2b2  in the  numerator and 4 in the denominator.  In the above problem first the expression was simplified then it was multiplied.  It can also be done vice versa.  First the rational-expressions can be  multiplied  that is numerator with numerator 4xy^2  with 2x and denominator with denominator 3y with 4y.
Then it can  be simplified.

4xy^2         2x
x
3y           4y

4x y^2. 2 x      =                4 . 2x^2 y^2
3y. .4y                             4. 3 y^2

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When multiplying the numerator we get 8x^2 y^2   and when multiplying the denominator we get 12 y^2


4.2 x^2 y^2

4.3y^2

The 4 in the numerator and denominator gets cancelled. The y^2 in the numerator and denominator also gets cancelled. And we are left out with

2 x^2
3

Which gives the final answer.

Practice Percentages

Introduction for practice percentage:

The word “percent” is consequent from Latin. It was at first “per centum”, which means “by the hundred”. Thus the statement is frequently complete that “percent means hundredths”.

Percentage deals with the collection of decimal fraction whose denominators are 100 – that is, fractions of two decimal spaces Since hundredths be used so regularly, the decimal position was drop and the symbol % be located after the number and understand “percent”. Thus, 0.25 and 25% represent the same value, 25/100.  The first is read “25 hundredths”, and the second is read “25 percent”. Both mean 25 parts out of 100.

Originally, percent is used in discussing relative values. For example, 25 percent may convey an idea of relative value or relationship. To say “35 percent of the crew is ashore” gives an idea of what part of the crew is gone, but it does not tell how many of discussing the percentage.

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Practice percentage for steps and example problems:


Steps for practice percentage:

STEP 1: Start the percentage x/100 = is/of. X is the percentage (over 100 of course), "is" refers to fraction, and "of" refers to entire.

STEP 2: In the question "80 is to 40 percent of what number? x=40, is=40 ("80 is"), and of = the unknown ("of what number"). Therefore write 40/100=80/x.

STEP 3: Cross multiply. You will have a constant value on one side and multiply a variable on the other side. Here it is 40x=8,000.

STEP 4: Solve for x. Here, x = 8,000/40 = 400, So x = 400.

Example problems for percentage:

Example problem 1:

The $169.99 poodle I purchased was on sale for 10% off, what did I pay for my poodle?

Solution:

$ 169.99 x 10 / 100

= $ 169.99 / 10

= $16.999

So Pay for poodle = $169.99 - $16.99

Answer = $153

Example problem 2:

I got 40% off when I purchased a rare comic book regularly priced at $74.50. How much did I pay?

Solution:

$74.50 x (40 / 100)

= $74.50 x (4 / 10)

= ($74.50 x 4) / 10

= $298 / 10 = $29.8

Pay for books = $74.50 - $29.8

= $44.7

Example problem 3:

Our take out dinner was $74.95 but we got 20% off because we picked it up. What did our meal end up costing us? $59.96

Solution:

$74.95 * (20 / 100)

= $74.95 * (2 / 10)

= ($74.95 * 2) / 10

=$149.9 /10 = $14.99

End of meal cost = $74.95 - $14.99

Answer = $ 59.96


Practice problem for percentage:


Practice problem 1:

My flight was $199.99 but I got 30% off because the plane wasn’t full. What did I pay?

Answer: $139.99

Practice problem 2:

$40.50 video games were on sale for 30% off, how much are they now?

Answer: $28.25

Practice problem 3:

My new cell phone cost me $190.00 but when I signed a 2 year plan, I got 20% off so it only cost me?

Answer: $152.00

Practice Problem 4:

It’s usually $16.50 to go to the show, but if you go on Tuesday, there’s a 40% discount, what does it cost to go to the show on Tuesday?

Answer: $9.90

Place Value Practice

Introduction for place value practice:

Place value is a second important concept in mathematics. It refers to the value of the digit, relative to its location in the number. The place value can be identified for any given digit in a written number, in the number 7403, the digit 4 is in the hundreds place and stands for 400. The 3 in 34 is not the same as the 3 in 23, even though both digits are 3. Is this topic Statistics Problems hard for you? Watch out for my coming posts.

This is because the 3 in 34 is in the tens place and therefore worth 30, while the 3 in 23 is in the ones place and worth only 3. Child is developing an understanding of place value up to at least the thousands. (This also means that she needs to practice skip counting starting with any number up to one thousand: 326,336,346)

Example: The number 3725 is read as three thousand seven hundred twenty- five. In it, there are 3 thousands, 7 hundreds, 2 tens, and 5 ones. A number with only ones has only one digit, one with tens two digits, and one with thousands has four digits.

Example problem for place value practice:

5 ones =             5

2 tens =            20

7 hundreds =     700

3 thousands =  3000

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Example problem for place value practice:


Read the numbers, and then tell the place and value of the underlined digit; Up to 10 millions.

1. 5,760,000

What place is the underlined digit in?

Ten thousands place

What is the value of the underlined digit?

60,000

2.   86,003,040

What place is the underlined digit in?

Millions place

What is the value of the underlined digit?

6,000,000

3.   4,954,560

What place is the underlined digit in?

Tens place

What is the value of the underlined digit?

60

4.    5,520,254

What place is the underlined digit in?

Millions place

What is the value of the underlined digit?

5,000,000