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Probability
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__FORCETOC__

__FORCETOC__

+= Introduction =

+A brief history of how probability was developed

+within the discipline of mathematics. Random processes can be

+modelled or explained mathematically by using a probability model.

+The two probability models are a) Experimental approach to

+probability b) Theoretical approach to probability. The basic

+principle of counting is covered.

+In everyday life, we come across statements such as

+ 1. It will probably rain today.

+ 2. I doubt that he will pass the test.

+ 3. Most probably, Kavita will stand first in the annual examination.

+ 4. Chances are high that the prices of diesel will go up.

+ 5. There is a 50-50 chance of India winning a toss in today’s match.

+
+The

+words '''‘probably’,'''

+‘doubt’, ‘most probably’, ‘chances’''','''

+etc., used in the statements above involve an element of uncertainty.

+For example, in (1), ‘probably rain’ will mean it may rain or may

+not rain today. We are predicting rain today based on our past

+experience when it rained under similar conditions. Similar

+predictions are also made in other cases listed in (2) to (5).

+
+The uncertainty of ‘probably’ etc. can be measured numerically by

+means of ‘probability’ in many cases. Though probability started

+with gambling, it has been used extensively in the fields of Physical

+Sciences, Commerce, Biological Sciences, Medical Sciences, WeatherForecasting,etc.

+
+Probability theory like many other branches of mathematics, evolved out of

+practical consideration. It had its origin in the 16th century when

+an Italian physician and mathematician Jerome Cardan (1501–1576)

+wrote the first book on the subject “Book on Games of Chance”

+(Biber de Ludo Aleae). It was published in 1663 after his death.

+
+When something occurs it is called an '''event'''.

+For example : A spinner has 4 equal sectors coloured

+yellow, blue, green and red. What are the chances of landing on blue

+after spinning the spinner? What are the chances of landing on red?

+The chances of landing on blue are 1 in 4, or one fourth. The chances

+of landing on red are 1 in 4, or one fourth.

+
+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.

+
+'''Impossible'''

+Event '''is'''

+an event that can never occur. The probability of landing on purple

+after spinning the spinner is impossible as it is

+impossible to land on purple since the spinner does not contain this

+colour.

+
+'''Certain'''

+events:

+That the event will surely occur. If we consider the situation where

+A

+teacher chooses a student at random from a class of 30 girls. What is

+the probability that the student chosen is a girl? Since all the

+students in the class are girls, the teacher is certain to choose a

+girl.

+
+== Historical Note ==

+In 1654, a gambler Chevalier de Metre approached the well known French

+Philosoher and Mathematician Blaise Pascal (1623–1662) for certain

+dice problem. Pascal became interested in these problems and

+discussed with famous French Mathematician Pierre de Fermat

+(1601–1665). Both Pascal and Fermat solved the problem

+independently. Besides, Pascal and Fermat, outstanding contributions

+to probability theory were also made by Christian Huygenes

+(1629–1665), a Dutchman, J. Bernoulli (1654–1705), De Moivre

+(1667–1754), a Frenchman Pierre Laplace (1749–1827), A Frenchman

+and the Russian P.L Chebyshev (1821–1897), A. A Markov (1856–1922)

+and A. N Kolmogorove (1903–1987). Kolmogorove is credited with the

+axiomatic theory of probability. His book ‘Foundations of

+Probability’ published in 1933, introduces probability as a set

+function and is considered a classic.

+
+== Experimental & Theoretical Approach ==

+A

+'''Random Experiment''' is an experiment, trial, or observation

+that can be repeated numerous times under the '''''same conditions'''''.

+The outcome of an individual random experiment must be independent

+and identically distributed. It must in no way be affected by any

+previous outcome and cannot be predicted with certainty.

+
+Examples of a Random experiment include:

+
+The tossing of a coin. The experiment can yield two possible outcomes,

+heads or tails.

+
+The roll of a die. The experiment can yield six possible outcomes, this

+outcome is the number 1 to 6 as the die faces are labelled.

+
+A complete list of all possible outcomes of a random experiment is

+called '''''sample space''''' or possibility space and is denoted by S

+
+In the coin tossing activity S = {heads, tails} and in the dice throwing

+activity S = {1,2,3,4,5,6}.

+
+Suppose we toss a coin in the air and note down the result each time. If we

+repeat this exercise say 10 times and note down the result each

+time. Each toss of a coin is called a '''trial'''.

+
+So, a trial is an action which results in one or several outcomes. The

+possible '''outcomes''' when we toss a coin are Head and Tail. Getting a head in a

+particular trial is an '''event''' with a particular outcome head.

+
+Now if we say let n be the number of trials, then the '''experimental'''

+probability P(E)''' of an event E happening is given by'''

+
+[[Image:KOER%20Probability,%20Permutations%20and%20Combinations_html_68e91ef4.gif]]

+
+The probability of E an event happening is always between 0 and 1 including 0 and 1,

+where 0 means it is impossible for the event to occur and 1 means its certain to occur.

+The

+'''theoretical'''

+probability

+(also called classical probability) of an event E, written as P(E),

+where we assume that the outcome of the events are ''equally''

+likely

+
+[[Image:KOER%20Probability,%20Permutations%20and%20Combinations_html_48cf88f6.gif]]

+
+In the case of the coin tossing ,

+
+[[Image:KOER%20Probability,%20Permutations%20and%20Combinations_html_m7f38b0db.gif]]

+
+'''Experimental probability'''

+
+The chances of something happening, based on

+repeated testing and observing results. It is the ratio of the number

+of times an event occurred to the number of times tested. For

+example, to find the experimental probability of winning a game, one

+must play the game many times, then divide the number of games won by

+the total number of games played '<nowiki/>''P'<nowiki/>'''''robability'''

+
+The measure of how likely it is for an event to

+occur. The probability of an event is always a number between zero

+and 100%. The meaning (interpretation) of probability is the subject

+of theories of probability. However, any rule for assigning

+probabilities to events has to satisfy the axioms of probability

+
+'''Random number generators'''

+
+A device used to produce a selection of numbers in

+a fair manner, in no particular order and with no favour being given

+to any numbers. Examples include dice, spinners, coins, and computer

+programs designed to randomly pick numbers

+
+'''Theoretical probability'''

+
+The chances of events happening as determined by

+calculating results that would occur under ideal circumstances. For

+example, the theoretical probability of rolling a 4 on a four-sided

+die is 1/4 or 25%, because there is one chance in four to roll a 4,

+and under ideal circumstances one out of every four rolls would be a

+4. Contrast with experimental probability

= Textbook =

= Textbook =

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= Teaching Outlines =

= Teaching Outlines =

−==Concept ~~#~~1 ~~Introduction to ~~Probability==

+==Concept - 1 Experimental Probability==

+=== Learning objectives ===

+Perform a random experiment and tabulate results and calculate the experimental probability of some events.

+=== Notes for teachers ===

+=== Activities ===

+# Activity No 1: [[Experimental Probability Activity 1|Experimental_Probability_Activity1]]

+# Activity No 2: [[Even and Odd Probability Activity2]]

+== Concept - 2 Introduction to Probability ==

−===Learning objectives===

+=== Learning objectives ===

#Understand that events occur with different frequencies

#Understand that events occur with different frequencies

#Different events have different likelihoods (likely, unlikely, equally likely, not equally likely)

#Different events have different likelihoods (likely, unlikely, equally likely, not equally likely)

#Understand the idea of sample space and universe of events

#Understand the idea of sample space and universe of events

−===Notes for teachers===

===Notes for teachers===

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#Compare the results across groups.

#Compare the results across groups.

#To develop an understanding of what chance means?

#To develop an understanding of what chance means?

−===Activities===

===Activities===

#Activity No #1 '''[[probability_introduction_activity1]]'''

#Activity No #1 '''[[probability_introduction_activity1]]'''

#Activity No #2 '''[[probability_introduction_activity2]]'''

#Activity No #2 '''[[probability_introduction_activity2]]'''

−==Concept #2 Different types of events==

==Concept #2 Different types of events==

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===Notes for teachers===

===Notes for teachers===

−===Activities===

===Activities===

#Activity No #1 '''[[probability_types_of_events_activity1]]'''

#Activity No #1 '''[[probability_types_of_events_activity1]]'''

−#Activity No #2 ~~'''[[probability_types_of_events_activity2]]'''~~

+#Activity No #2

==Concept #3 Conditional probability==

==Concept #3 Conditional probability==

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#Activity No #2 '''Concept Name - Activity No.'''

#Activity No #2 '''Concept Name - Activity No.'''

−== Further Explorations ==

+#Math Probability - What a Fun Unit!, http://www.algebra-class.com/math-probability.html

+#Khan Academy Probability Part1, []

+#Khan Academy Probability Part1, []

+#Lecture - 1 Introduction to the Theory of Probability, http://www.youtube.com/watch?v=r1sLCDA-kNY&feature=related

=Assessment activities for CCE=

=Assessment activities for CCE=

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= Math Fun =

= Math Fun =

+
+[[Category:Class 10]]

+[[Category:Probability]]