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__FORCETOC__

__FORCETOC__
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= Introduction =
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A brief history of how probability was developed
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within the discipline of mathematics. Random processes can be
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modelled or explained mathematically by using a probability model.
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The two probability models are a) Experimental approach to
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probability b) Theoretical approach to probability. The basic
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principle of counting is covered.
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In everyday life, we come across statements such as
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1. It will probably rain today.
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2. I doubt that he will pass the test.
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3. Most probably, Kavita will stand first in the annual examination.
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4. Chances are high that the prices of diesel will go up.
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5. There is a 50-50 chance of India winning a toss in today’s match.
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The
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words '''‘probably’,'''
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‘doubt’, ‘most probably’, ‘chances’''','''
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etc., used in the statements above involve an element of uncertainty.
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For example, in (1), ‘probably rain’ will mean it may rain or may
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not rain today. We are predicting rain today based on our past
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experience when it rained under similar conditions. Similar
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predictions are also made in other cases listed in (2) to (5).
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The uncertainty of ‘probably’ etc. can be measured numerically by
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means of ‘probability’ in many cases. Though probability started
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with gambling, it has been used extensively in the fields of Physical
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Sciences, Commerce, Biological Sciences, Medical Sciences, WeatherForecasting,etc.
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Probability theory like many other branches of mathematics, evolved out of
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practical consideration. It had its origin in the 16th century when
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an Italian physician and mathematician Jerome Cardan (1501–1576)
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wrote the first book on the subject “Book on Games of Chance”
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(Biber de Ludo Aleae). It was published in 1663 after his death.
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When something occurs it is called an '''event'''.
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For example : A spinner has 4 equal sectors coloured
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yellow, blue, green and red. What are the chances of landing on blue
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after spinning the spinner? What are the chances of landing on red?
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The chances of landing on blue are 1 in 4, or one fourth. The chances
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of landing on red are 1 in 4, or one fourth.
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An
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'''experiment'''
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is a situation involving chance or probability that leads to results
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called outcomes. In the problem above, the experiment is spinning the
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spinner.
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An
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'''outcome'''
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is the result of a single trial of an experiment. The possible
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outcomes are landing on yellow, blue, green or red.
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An
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'''event'''
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is one or more outcomes of an experiment. One event of this
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experiment is landing on blue.
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'''Probability'''
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is the measure of how likely an event is. The probability of landing
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on blue is one fourth.
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'''Impossible'''
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Event '''is'''
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an event that can never occur. The probability of landing on purple
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after spinning the spinner is impossible as it is
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impossible to land on purple since the spinner does not contain this
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colour.
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'''Certain'''
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events:
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That the event will surely occur. If we consider the situation where
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A
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teacher chooses a student at random from a class of 30 girls. What is
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the probability that the student chosen is a girl? Since all the
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students in the class are girls, the teacher is certain to choose a
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girl.
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== Historical Note ==
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In 1654, a gambler Chevalier de Metre approached the well known French
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Philosoher and Mathematician Blaise Pascal (1623–1662) for certain
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dice problem. Pascal became interested in these problems and
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discussed with famous French Mathematician Pierre de Fermat
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(1601–1665). Both Pascal and Fermat solved the problem
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independently. Besides, Pascal and Fermat, outstanding contributions
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to probability theory were also made by Christian Huygenes
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(1629–1665), a Dutchman, J. Bernoulli (1654–1705), De Moivre
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(1667–1754), a Frenchman Pierre Laplace (1749–1827), A Frenchman
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and the Russian P.L Chebyshev (1821–1897), A. A Markov (1856–1922)
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and A. N Kolmogorove (1903–1987). Kolmogorove is credited with the
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axiomatic theory of probability. His book ‘Foundations of
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Probability’ published in 1933, introduces probability as a set
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function and is considered a classic.
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== Experimental & Theoretical Approach ==
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A
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'''Random Experiment''' is an experiment, trial, or observation
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that can be repeated numerous times under the '''''same conditions'''''.
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The outcome of an individual random experiment must be independent
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and identically distributed. It must in no way be affected by any
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previous outcome and cannot be predicted with certainty.
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Examples of a Random experiment include:
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The tossing of a coin. The experiment can yield two possible outcomes,
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+
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The roll of a die. The experiment can yield six possible outcomes, this
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outcome is the number 1 to 6 as the die faces are labelled.
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A complete list of all possible outcomes of a random experiment is
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called '''''sample space''''' or possibility space and is denoted by S
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In the coin tossing activity S = {heads, tails} and in the dice throwing
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activity S = {1,2,3,4,5,6}.
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Suppose we toss a coin in the air and note down the result each time. If we
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repeat this exercise say 10 times and note down the result each
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time. Each toss of a coin is called a '''trial'''.
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So, a trial is an action which results in one or several outcomes. The
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possible '''outcomes''' when we toss a coin are Head and Tail. Getting a head in a
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particular trial is an '''event''' with a particular outcome head.
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Now if we say let n be the number of trials, then the '''experimental'''
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probability P(E)''' of an event E happening is given by'''
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[[Image:KOER%20Probability,%20Permutations%20and%20Combinations_html_68e91ef4.gif]]
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The probability of E an event happening is always between 0 and 1 including 0 and 1,
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where 0 means it is impossible for the event to occur and 1 means its certain to occur.
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The
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'''theoretical'''
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probability
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(also called classical probability) of an event E, written as P(E),
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where we assume that the outcome of the events are ''equally''
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likely
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[[Image:KOER%20Probability,%20Permutations%20and%20Combinations_html_48cf88f6.gif]]
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In the case of the coin tossing ,
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[[Image:KOER%20Probability,%20Permutations%20and%20Combinations_html_m7f38b0db.gif]]
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'''Experimental probability'''
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The chances of something happening, based on
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repeated testing and observing results. It is the ratio of the number
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of times an event occurred to the number of times tested. For
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example, to find the experimental probability of winning a game, one
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must play the game many times, then divide the number of games won by
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the total number of games played '<nowiki/>''P'<nowiki/>'''''robability'''
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The measure of how likely it is for an event to
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occur. The probability of an event is always a number between zero
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and 100%. The meaning (interpretation) of probability is the subject
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of theories of probability. However, any rule for assigning
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probabilities to events has to satisfy the axioms of probability
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'''Random number generators'''
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A device used to produce a selection of numbers in
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a fair manner, in no particular order and with no favour being given
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to any numbers. Examples include dice, spinners, coins, and computer
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programs designed to randomly pick numbers
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'''Theoretical probability'''
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The chances of events happening as determined by
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calculating results that would occur under ideal circumstances. For
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example, the theoretical probability of rolling a 4 on a four-sided
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die is 1/4 or 25%, because there is one chance in four to roll a 4,
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and under ideal circumstances one out of every four rolls would be a
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4. Contrast with experimental probability

= Textbook =

= Textbook =
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= Teaching Outlines =

= Teaching Outlines =
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==Concept #1 Introduction to Probability==
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==Concept - 1 Experimental Probability==
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=== Learning objectives ===
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Perform a random experiment and tabulate results and calculate the experimental probability of some events.
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=== Notes for teachers ===
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=== Activities ===
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# Activity No 1: [[Experimental Probability Activity 1|Experimental_Probability_Activity1]]
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# Activity No 2: [[Even and Odd Probability Activity2]]
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== Concept - 2 Introduction to Probability ==
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===Learning objectives===
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=== 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]]'''
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#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.'''
−

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== Further Explorations ==
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#Math  Probability - What a Fun Unit!,  http://www.algebra-class.com/math-probability.html
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#Khan Academy Probability Part1, []
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#Khan Academy Probability Part1, []
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#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 =
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[[Category:Class 10]]
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[[Category:Probability]]
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