Probability

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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

KOER Probability, Permutations and Combinations 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

KOER Probability, Permutations and Combinations html 48cf88f6.gif

In the case of the coin tossing ,

KOER Probability, Permutations and Combinations 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 'P'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

Please click here for Karnataka and other text books.
Karnataka text book for Class 10, Chapter 05-Probability

Additional Information


Useful websites

  1. To get the information about probability click here

Lessons and activities:

http://www-tc.pbs.org/teachers/mathline/lessonplans/pdf/esmp/chancesare.pdf http://www.bbc.co.uk/schools/teachers/ks2_lessonplans/maths/probability.shtml

Activities on Probability:

http://www.cimt.plymouth.ac.uk/projects/mepres/allgcse/as5act1.pdf

How to teach probability

http://nrich.maths.org/probability http://nrich.maths.org/9853

Possible video resource for dubbing

  1. https://www.youtube.com/watch?v=QpfMwA0z_1Y
  2. https://youtu.be/Ki0ROFD8MzU

Reference Books

  1. http://www.teachingideas.co.uk/maths/probabilitycards.htm
  2. http://www.teachingideas.co.uk/maths/files/probabilitycards.pdf

Teaching Outlines

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

  1. Activity No 1: Experimental_Probability_Activity1
  2. Activity No 2: Even and Odd Probability Activity2

Concept - 2 Introduction to Probability

Learning objectives

  1. Understand that events occur with different frequencies
  2. Different events have different likelihoods (likely, unlikely, equally likely, not equally likely)
  3. Understand the idea of sample space and universe of events

Notes for teachers

  1. To understand likelihood of events happening
  2. The objective here is not numerical computation but to understand events, likelihoods and vocabulary of description. The activity can be done in groups/ pairs. Not a whole class activity.
  3. Compare the results across groups.
  4. To develop an understanding of what chance means?

Activities

  1. Activity No #1 probability_introduction_activity1
  2. Activity No #2 probability_introduction_activity2

Concept #2 Different types of events

Learning objectives

  1. Understand elementary and compound events and construction of such events
    1. Complementary events
    2. Independent events / Mutually exclusive events

Notes for teachers

Activities

  1. Activity No #1 probability_types_of_events_activity1
  2. Activity No #2

Concept #3 Conditional probability

Learning objectives

Notes for teachers

Activities

  1. Activity No #1 Concept Name - Activity No.
  2. Activity No #2 Concept Name - Activity No.

Further Explorations

  1. Math Probability - What a Fun Unit!, http://www.algebra-class.com/math-probability.html
  2. Khan Academy Probability Part1, []
  3. Khan Academy Probability Part1, []
  4. Lecture - 1 Introduction to the Theory of Probability, http://www.youtube.com/watch?v=r1sLCDA-kNY&feature=related

Assessment activities for CCE

Hints for difficult problems

Project Ideas

Math Fun