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| = Concept Map = | | = Concept Map = |
− | <mm>[[probability .mm|Flash]]</mm>
| + | [[File:probability.mm|Flash]] |
| | | |
| __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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| [http://ktbs.kar.nic.in/New/Textbooks/class-x/english/maths/class-x-english-maths-chapter05.pdf Karnataka text book for Class 10, Chapter 05-Probability] | | [http://ktbs.kar.nic.in/New/Textbooks/class-x/english/maths/class-x-english-maths-chapter05.pdf Karnataka text book for Class 10, Chapter 05-Probability] |
| | | |
− | =Additional Information= | + | =Additional Information= |
| + | |
| + | {{#widget:YouTube|id=rlUZXrJGuf8}} {{#widget:YouTube|id=ihH7ZXemEdI}} <br> |
| ==Useful websites== | | ==Useful websites== |
| + | # To get the information about probability [https://www.mathsisfun.com/data/probability.html click here] |
| + | |
| ===Lessons and activities:=== | | ===Lessons and activities:=== |
| http://www-tc.pbs.org/teachers/mathline/lessonplans/pdf/esmp/chancesare.pdf | | http://www-tc.pbs.org/teachers/mathline/lessonplans/pdf/esmp/chancesare.pdf |
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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 === |
| | | |
− | ===Learning objectives=== | + | === 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 === |
| #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 |
− |
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− |
| |
| | | |
| ===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]]''' |
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− |
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| | | |
| ==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]] |