**Probability theory:** In many real life situations when we are unable to forecast the future with complete certainty. That is, in many decisions, we face the uncertainty. This leads to the study and use of the probability theory.

## Introduction

The first attempt to give quantitative measure of probability was made by Galileo (1564-1642), he was an Italian mathematician.

The first contribution was given by the two mathematicians Pascal and Fermat due to a gambler’s dispute in 1654 which led to the creation of a mathematical theory of sprobability by them.

Later, important contributions were made by various researchers including Huyghens, Jacob Bernoulli (1654-1705), Laplace (1749-1827), Abraham De-Moivre (1667-1754), and Markov (1856-1922).

Thomas Bayes gave an important technical result known as Bayes’ theorem, published after his death in 1763, using which probabilities can be revised on the basis of some new information.

**Random Experiment**

An experiment in which we know all the possible outcomes but we can not say that or we can not predict that which of them will occur when we perform the experiment.

Suppose we toss a coin, we know that what are possible outcomes, it would be head or tail, but we do not know which one of these two will occur.

So tossing a coin is a random experiment.

Similarly, ‘Throwing a die’ and ‘Drawing a card from a well shuffled pack of 52 playing cards ‘are the examples of random experiment.

**Trial**

We Perform an experiment which we call it a trial, For example-

(i) Throwing a dice

(ii) Tossing a coin

(iii) Picking a playing card

**Important definitions**

**1. Die: **It is a small cube. There are 6 numbers on faces- 1,2,3,4,5,6 A Plural of the die is dice. On throwing a die, the outcome is the number of dots on its upper face..

**2. Cards: **A pack of cards consists of four suits i.e. Spades, Hearts, Diamonds and Clubs. Each suit consists of 13 cards, nine cards numbered 2, 3, 4, …, 10, an Ace, a King, a Queen and a Jack or Knave.

Colour of Spades and Clubs is black and that of Hearts and Diamonds is red. Kings, Queens and Jacks are known as **face **cards.

A *probability space* is a three-tuple (S, F, P) in which the three components are

**1.** **Sample space:** A non-empty set S called the *sample space*, which represents all possible outcomes.

**2.** **Event space:** A collection F of subsets of S, called the *event space*.

**3.** **Probability function:** A function P : F tends to R, that assigns probabilities to the events in F.

**Odds in favour of an event and odds against an event**

If number of favourable ways = m, number of not favourable events = n

(i) Odds in favour of the event =m/n

(ii) Odds against the event =n/m

**Sample Space and Discrete Sample Space**

**Sample space**

Set of all possible outcomes of a random experiment is known as sample space and we denote it by S, and the total number of elements in the sample space is known as size of the sample space and is denoted by n(S).

**Discrete sample space- **Sample space in which sample points are finite or countably infinite is called discrete sample space.

For example-

1. If a die is thrown, then the sample space is

S = {1, 2, 3, 4, 5, 6} and n(S) = 6.

2. If a coin is tossed twice or two coins are tossed simultaneously then the sample space is

S = {HH, HT, TH, TT}

3. If a coin is tossed 4 times or four coins are tossed simultaneously then the sample space is

S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT,

THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT} and n(S) = 16.

**Note-If a random experiment with x possible outcomes is performed n times, then the total number of elements in the sample is x^n.**

**Events and their types**

1. Exhaustive Events or Sample Space: The set of all possible outcomes of a single performance of an experiment is exhaustive events or sample space. Each outcome is called a sample point.

3. Equally likely events: Two events are said to be ‘equally likely’, if one of them cannot be expected in preference to the other. For instance, if we draw a card from well-shuffled pack, we may get any card. Then the 52 different cases are equally likely.

4. Independent events: Two events may be independent, when the actual happening of one does not influence in any way the probability of the happening of the other.

5. Mutually Exclusive events: Two events are known as mutually exclusive, when the occurrence of one of them excludes the occurrence of the other. For example, on tossing of a coin, either we get head or tail, but not both.

6. Compound Event: When two or more events occur in composition with each other, the simultaneous occurrence is called a compound event. When a die is thrown, getting a 5 or 6 is a compound event.

7. Favourable Events: The events, which ensure the required happening, are said to be favourable events. For example, in throwing a die, to have the even numbers, 2, 4 and 6 are favourable cases.

**Algebra of Events**

In a random experiment, let S be the sample space.

Let and be the events, then we say that-

1. is an event that occurs only when both A and B occurs.

1. (A∪B) is an event that occurs when either one of A or B occurs.

A’ is an event that occurs only when A does not occurs.

Let S be the sample space of a random experiment and events A and B S then

For three non-mutually exclusive events A, B and C, we have

NOTE- If events A and B are mutually exclusive events, then-

**P(A U B) = P(A) + P(B)**

For three mutually exclusive events A, B and C, we have

**Example: 25 lottery tickets are marked with first 25 numerals. A ticket is drawn at random. Find the probability that it is a multiple of 5 or 7.**

Sol.

Let A be the event that the drawn ticket bears a number multiple of 5 and B be the event that it bears a number multiple of 7.

Therefore,

A = {5, 10, 15, 20, 25}

B = {7, 14, 21}

Here, as

Therefore, A and B are mutually exclusive, and hence

**Example: A Card is drawn from a pack of 52 playing cards, find the proabability that the drawn card is an ace or a red colour card.**

Sol.

Let A be the event that the drawn card is a card of ace and B be the event that it is red colour card. Now as there are four cards of ace and 26 red colour cards in a pack of 52 playing cards. Also, 2 cards in the pack are ace cards of red colour.

**Mutually Exclusive and Exhaustive Events, Complimentary events**

**Exhaustive Cases-**

The total number of possible outcomes in a random experiment is called the exhaustive cases.

Or

The number of elements in the sample space is known as number of exhaustive cases

**For example:**

1. When we toss a coin, then the number of exhaustive cases is 2 and the sample space in this case is {H, T}.

2. When we throw a die then number of exhaustive cases is 6 and the sample space in this case is {1, 2, 3, 4, 5, 6}

**Mutually Exclusive Cases**

If the happening of any one of them prevents the happening of all others in a single experiment, then this case is said to be mutually exclusive.

For example:

In throwing a dice experiment all 1, 2, 3, 4, 5, 6 are mutually exclusive as there cannot be simultaneous occurrence 1, 2, 3, 4, 5, 6.

**Complementary events**

Two events are said to be complementary events when one event occur iff other does not.

The possibility of two complementary events adds up to 1.

The complement of A is – A’

**Classical definition of Probability**

Suppose there are ‘n’ exhaustive cases in a random experiment which is equally likely and mutually exclusive.

Let ‘m’ cases are favourable for the happening of an event A, then the probability of happening event A can be defined as-

Probability of non-happening of the event A is-

**Note-** Always remember that the probability of any events lies between 0 and 1.

**Note- Probability of an impossible event is always zero and that of certain event is 1.**

**Expected value**

Let are the probabilities of events and respectively. Then we can define the expected value as below-

## Solved example

**Example: A bag contains 7 red and 8 black balls then find the probability of getting a red ball.**

Sol.

Here total cases = 7 + 8 = 15

According to the definition of probability,

So that, here favourable cases- red balls = 7

Then,

**Example: A bag contains 4 red, 5 black and 2 green balls. i**f someone draw o**ne ball from the bag. Find the probability that-**

**(i) It is a red ball**

**(ii) It is not black**

**(iii) It is green or black**

Sol.

There are total 11 exhaustive cases.

1. We know that, by the definition

And the favourable cases are 4, then the probability of getting a red ball is-

Similarly-

Probability of getting a ball which is not black is-

Probability of getting a green or black ball is-

**Example: If I throw a fair die then find the probability of getting**

**(i) A prime number**

**(ii) An even number**

**(iii) A number multiple of 2 or 3**

Sol:

The sample space in this case is- S = {1, 2, 3, 4, 5, 6}

(i) Let E1 be the event of getting a prime number, then E1 = {2, 3, 5}.

P(E1) = 3/6 or ½

(ii) Let E2 be the event of getting an even number, then

E2 = {2, 4, 6}.

P(E2) = 3/6 or ½

(iii) Let E3 be the event of getting an even number, then

E3 = {2, 3, 4, 6 }

P(E3) = 4/6 or 2/3

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