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CBSE Class 12 Commerce Mathematics Probability Introduction

CBSE Class 12 Commerce Mathematics Probability Introduction

CBSE Class 12 Commerce Mathematics Probability : The Central Board of Secondary Education (abbreviated as CBSE) is a Board of Education for public and private schools, under the Union Government of India. Central Board of Secondary Education (CBSE) has asked all schools affiliated to follow only NCERT curriculum.

The first education board to be set up in India was the Uttar Pradesh Board of High School and Intermediate Education in 1921, which was under jurisdiction of Rajputana, Central India and Gwalior. In 1929, the government of India set up a joint Board named “Board of High School and Intermediate Education, Rajputana”. This included Ajmer, Merwara, Central India and Gwalior. Later it was confined to Ajmer, Bhopal and Vindhya Pradesh. In 1952, it became the “Central Board of Secondary Education”.

CBSE Class 12 Commerce Mathematics Probability Complete Notes

CBSE Class 12 Commerce Mathematics Probability :  Cakart team members provides here CBSE Class 12 Commerce Mathematics Probability Complete Notes and other CBSE Class 12 Commerce Mathematics Complete Notes in pdf format. We provides you direct link for downloading CBSE Class 12 Commerce Mathematics Probability Complete Notes in pdf format. Download CBSE Class 12 Commerce Mathematics Probability Complete Notes and read well.

CBSE Class 12 Commerce Mathematics Probability Complete Notes

CBSE Class 12 Commerce Mathematics Probability : Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means “that which is learnt”,”what one gets to know”, hence also “study” and “science”, and in modern Greek just “lesson”. The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean “to learn”. In Greece, the word for “mathematics” came to have the narrower and more technical meaning “mathematical study” even in Classical times. Its adjective is μαθηματικός (mathēmatikós), meaning “related to learning” or “studious”, which likewise further came to mean “mathematical”. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant “the mathematical art”.

Download here CBSE Class 12 Commerce Mathematics Probability Complete Notes in pdf format

CBSE Class 12 Commerce Mathematics Probability Complete Notes

CBSE Class 12 Commerce Mathematics Probability : American Heritage Dictionary defines Probability Theory as the branch of Mathematics that studies the likelihood of occurrence of random events in order to predict the behavior of defined systems. (Of course What Is Random? is a question that is not all that simple to answer.)

Starting with this definition, it would (probably 🙂 be right to conclude that the Probability Theory, being a branch of Mathematics, is an exact, deductive science that studies uncertain quantities related to random events. This might seem to be a strange marriage of mathematical certainty and uncertainty of randomness. On a second thought, though, most people will agree that a newly conceived baby has a 50-50 chance (exact but, likely, inaccurate estimate) to be, for example, a girl or a boy, for that matter.

Interestingly, a recent book by Marilyn vos Savant dealing with people’s perception of probability and statistics is titled The Power of Logical Thinking. My first problems will be drawn from this book.

As with other mathematical problems, it’s often helpful to experiment with a problem in order to gain an insight as to what the correct answer might be. By necessity, probabilistic experiments require computer simulation of random events. It must sound as an oxymoron – a computer (i.e., deterministic device) producing random events – numbers, in our case, to be exact. See, if you can convince yourself that your computer can credibly handle this task also. A knowledgeable reader would, probably, note that this is a program (albeit deterministic) and not the computer that does the random number simulation. That’s right. It’s me and not your computer to blame if the simulation below does not exactly produce random numbers.

When you press the “Start” button below, the program will start random selection. Every second it will pick up one of the three numbers – 1, 2, or 3. You can terminate the process anytime by pressing the “Stop” button. Frequencies of selections appear in the corresponding input boxes. Do they look random?

123

Actually, the process of selection includes no selection at all. As a mathematician Robert Coveyou from the Oak Ridge National Laboratory has said, The generation of random numbers is too important to be left to chance. Instead, I have a function that is invoked every second. Each time it’s invoked, it produces one of the three 1, 2, 3 numbers. This is how the function works.

I start with an integer seed = 0. When a new random number is needed, the seed is replaced with the result of the following operation

seed = (7621 × seed + 1) mod 9999

In other words, in order to get a new value of seed, multiply the old value by 7621, add 1, and, finally, take the result modulo 9999. Now, assume, as in the example above, we need a random selection from the triple 1, 2, 3. That is, we seek a random integer n satisfying 1 ≤ n ≤ 3. The formula is

n = [3 × seed/9999] + 1.

Taking it step by step, dividing seed by 9999 produces a nonnegative real number between 0 and 1. This times 3 gives a real number between 0 and 3. Brackets reduce the latter to the nearest integer which is not greater than the number itself. The result is a nonnegative integer that is less than 3. Adding 1 makes it one of the three 1, 2, or 3.

CBSE Class 12 Commerce Mathematics Probability Complete Notes

CBSE Class 12 Commerce Mathematics Probability : These lessons on probability will include the following topics: Samples in probability, Probability of events, Theoretical probability, Experimental probability, Probability problems, Tree diagrams, Mutually exclusive events, Independent events, Dependent events, Factorial, Permutations, Combinations, Probability in Statistics, Probability and Combinatorics.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
This problem asked us to find some probabilities involving a spinner. Let’s look at some definitions and examples from the problem abov
DefinitionExample
An experiment is a situation involving chance or probability that leads to results called outcomes.In the problem above, the experiment is spinning the spinner.
An outcome is the result of a single trial of an experiment.The possible outcomes are landing on yellow, blue, green or red.
An event is one or more outcomes of an experiment.One event of this experiment is landing on blue.
Probability is the measure of how likely an event is.The probability of landing on blue is one fourth.

In order to measure probabilities, mathematicians have devised the following formula for finding the probability of an event

Probability Of An Event
P(A) =The Number Of Ways Event A Can Occur
The total number Of Possible Outcomes

The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. Let’s take a look at a slight modification of the problem from the top of the page.

Experiment 1:A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color? spinner
Outcomes:The possible outcomes of this experiment are yellow, blue, green, and red.
Probabilities:
P(yellow) =# of ways to land on yellow =1
total # of colors 4
P(blue) =# of ways to land on blue =1
total # of colors 4
P(green) =# of ways to land on green =1
total # of colors 4
P(red) =# of ways to land on red =1
total # of colors 4

 

Experiment 2:A single 6-sided die is rolled. What is the probability of each outcome? What is the probability of rolling an even number? of rolling an odd number?  dice


Outcomes:The possible outcomes of this experiment are 1, 2, 3, 4, 5 and 6.
Probabilities:
P(1) =# of ways to roll a 1 =1
total # of sides6
P(2) =# of ways to roll a 2 =1
total # of sides6
P(3) =# of ways to roll a 3 =1
total # of sides6
P(4) =# of ways to roll a 4 =1
total # of sides6
P(5) =# of ways to roll a 5 =1
total # of sides6
P(6) =# of ways to roll a 6 =1
total # of sides6
P(even) =# ways to roll an even number =3 =1
total # of sides62
P(odd) =# ways to roll an odd number =3 =1
total # of sides62

 

Experiment 2 illustrates the difference between an outcome and an event. A single outcome of this experiment is rolling a 1, or rolling a 2, or rolling a 3, etc. Rolling an even number (2, 4 or 6) is an event, and rolling an odd number (1, 3 or 5) is also an event.

 

In Experiment 1 the probability of each outcome is always the same. The probability of landing on each color of the spinner is always one fourth. In Experiment 2, the probability of rolling each number on the die is always one sixth. In both of these experiments, the outcomes are equally likely to occur. Let’s look at an experiment in which the outcomes are not equally likely.

 

Experiment 3:A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble? [IMAGE]
Outcomes:The possible outcomes of this experiment are red, green, blue and yellow.
Probabilities:
P(red) =# of ways to choose red = 6  = 3 
total # of marbles2211
P(green) =# of ways to choose green = 5 
total # of marbles22
P(blue) =# of ways to choose blue = 8  = 4 
total # of marbles2211
P(yellow) =# of ways to choose yellow = 3 
total # of marbles22

 

The outcomes in this experiment are not equally likely to occur. You are more likely to choose a blue marble than any other color. You are least likely to choose a yellow marble.

 

Experiment 4:Choose a number at random from 1 to 5. What is the probability of each outcome? What is the probability that the number chosen is even? What is the probability that the number chosen is odd?
Outcomes:The possible outcomes of this experiment are 1, 2, 3, 4 and 5.
Probabilities:
P(1) =# of ways to choose a 1 =1
total # of numbers5
P(2) =# of ways to choose a 2 =1
total # of numbers5
P(3) =# of ways to choose a 3 =1
total # of numbers5
P(4) =# of ways to choose a 4 =1
total # of numbers5
P(5) =# of ways to choose a 5 =1
total # of numbers5
P(even) =# of ways to choose an even number =2
total # of numbers5
P(odd) =# of ways to choose an odd number =3
total # of numbers5
The outcomes 1, 2, 3, 4 and 5 are equally likely to occur as a result of this experiment. However, the events even and odd are not equally likely to occur, since there are 3 odd numbers and only 2 even numbers from 1 to 5.
Summary:The probability of an event is the measure of the chance that the event will occur as a result of an experiment. The probability of an event A is the number of ways event A can occur divided by the total number of possible outcomes. The probability of an event A, symbolized by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event in the following way:

  • If P(A) > P(B) then event A is more likely to occur than event B.
  • If P(A) = P(B) then events A and B are equally likely to occur.

CBSE Class 12 Commerce Mathematics Probability Complete Notes

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