# 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.

# 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 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
**probability_types_of_events_activity2**

## Concept #3 Conditional probability

### Learning objectives

### Notes for teachers

### Activities

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