# Probability

Philosophy of Mathematics |

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# Concept Map

# 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**

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

In the case of the coin tossing ,

**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

- 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

## Reference Books

- http://www.teachingideas.co.uk/maths/probabilitycards.htm
- 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

- Activity No 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
- Different events have different likelihoods (likely, unlikely, equally likely, not equally likely)
- Understand the idea of sample space and universe of events

### Notes for teachers

- To understand likelihood of events happening
- 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.
- Compare the results across groups.
- To develop an understanding of what chance means?

### Activities

- Activity No #1
**probability_introduction_activity1** - Activity No #2
**probability_introduction_activity2**

## Concept #2 Different types of events

### Learning objectives

- Understand elementary and compound events and construction of such events
- Complementary events
- Independent events / Mutually exclusive events

### Notes for teachers

### Activities

- Activity No #1
**probability_types_of_events_activity1** - Activity No #2

## Concept #3 Conditional probability

### Learning objectives

### Notes for teachers

### Activities

- Activity No #1
**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