**PROBABILITY**

**1.1. DEFITITION**

**Random Experiments-**

For any invention, number of experiments are done. Consider an experiment whose results is not predictable under almost similar working condition then these experiments are known as Random Experiments**.**

These are some cases of random experiments-

**Case 1**: If we toss a coin, then the result of the experiment whether it is going to come head or tail is not predictable under very similar conditions.

**Case 2**: If we throw a dice, then the outcome of this can not be predicted with certainty that which number is going to turn.

**Sample Space,S –**

Each random experiments of some possible outcomes, if we make a set of all the possible outcomes of random experiments then Set ‘S’ is known as the Sample Space & each possible outcome is Sample Point.

**Case 1**: If we roll a die, then set of all possible outcomes, is given by {1, 2, 3, 4, 5, 6} then this will be the sample space of given experiment and 1, 2, 3, 4, 5 & 6 are sample points.

Similarly, if our objective is getting odd number on rolling same die then the Sample space will be {1, 3, 5} & for even number Sample space will be {2, 4, 6}.

**Case 2: **If the outcome of our experiment is to determination whether a male is married or not then our Sample space will be {Married, Unmarried}.

**Event,E**

An *ev*ent is a subset A of the sample space S, i.e., it is a set of possible outcomes.

An Event is a set of consisting some of the possible outcomes from the sample space of the experiment.

**Case 1:** On tossing a coin twice, All possible outcomes (Sample space) is {HH, HT, TH, TT} whereas {HH}{HH, TT},{HT, HH}, {HH, HT, TT} are the events.

If the event consists only single outcome then it is known as **Simple Events**.

If the events consist of more than one outcome then its is known as **Compound Events**.

**Types of Events- **

**(i)Complementary Event – **Any Event E^{C} is called complementary event of event E if it consists of all possible outcomes of sample space which is not present in E.

**Ex - **If we roll a die, then set of all possible outcomes, is given by {1, 2, 3, 4, 5, 6}.

An event of getting outcome in multiple of 3 is

E (multiples of 3) = {3,6}

Then, E^{C }= {1,2,4,5}

**(ii) Equally Likely Event – **if any two event of sample space are in such a way that the chance of both the events are equal, then this type of events is known as Equally likely events.

**Ex – **Chances of a new born baby to be a boy or girl is 50% means either it can be a girl or boy.

**(iii) Mutually Exclusive Events – **Two events are called as mutually exclusive when occurring of both the simultaneously is not possible.

If E_{1} & E_{2} are mutually exclusive then E_{1} ⋂ E_{2 }= ϕ

**Ex –** if we toss a coin then either head or tail can occur, occurrence of both simultaneously is not possible.

**(iv) Collectively Exhaustive Events - **Two events are called as Collectively exclusive when sample points of both the events incudes all the possible outcomes.

If E_{1} & E_{2} are mutually exclusive then E_{1 }⋃ E_{2 }= S

**Ex –** if we toss a coin & E_{1} is the occurrence of head and E_{2} is the occurrence of a tail. Then both the events are collectively exhaustive because both o them collectively include all possible outcomes.

**(v) Independent Events – **Two events are called as independent when occurring of 1^{st} event does not affect the occurrence of 2^{nd}.

**Ex – **On rolling two dice simultaneously, occurrence of 5 in 1^{st} die does not affect the occurrence of 4 in second die. Their occurrence is independent to each other.

**Probability –**If an experiment is conducted under essentially given condition upto ‘n’ times and let ‘m’ cases are favourable to an event ‘E’, then probability of ‘E’ is denoted by P(E) & defined as

**Example -1** A card is drawn from a deck of playing cards. What is the probability of that the card is

(i) Face card

(ii) Heart card

(iii) Face and heart card

**Sol. **

Total number of cards in a deck, n = 52 (sample space)

Total number of suits in a deck = 4(heart, spades, club, diamond)

Total face card (King, Queen, Jack) = 12(3 in each suit)

(i)Probability of card is face card

(ii) Probability of card is heart card

Number of heart card in a deck, m = 13

(iii) Probability of card to be face and heart

Number of face card with heart suit is, m = 3

**1.2. The Axioms of Probability**

Consider an Experiment whose sample space is S. For each event E of the sample space, we associate a real number P(E). Then P is called a probability function, and P(E) the probability of the event E, then P(E) will satisfies the following axioms.

**Axiom 1 **For every event E,

P(E) ≥ 0

Probability of an event can never be negative.

**Axiom 2 **In case of sure or certain event E,

P(E) = 1

Probability of an event with 100% surety is 1.

**Axiom 3 **For any number of** mutually exclusive** events E_{1}, E_{2}, ….,

P(E_{ 1}∪E_{ 2}∪E_{3}…) = P(E_{ 1}) + P(E_{2}) + p(E_{3}) …..

In particular, for two mutually exclusive events E_{1}, E_{2},

P(E_{ 1}∪E_{ 2}) = P(E_{ 1}) + P(E_{ 2})

**Example – 2 **A fair die is tossed once. Find the probability of a 2 or 5 turning up.

**Sol.**

When a fair die is rolled once, the sample space is S = {1, 2, 3, 4, 5, 6}.

Since die is fair thus, we assign equal probabilities to each sample points,

The event that either 2 or 5 turns up is indicated by (2 ∪ 5).

Therefore,

**1.3. Some Important Theorems on Probability**

From the above axioms we can now prove various theorems on probability

**Theorem 1:** For every event E,

0 ≤ P(E) ≤ 1,

i.e., a probability is between 0 and 1.

**Theorem 2:** P(Φ) = 0

i.e., the impossible event has probability zero.

**Theorem 3: ** If E^{C} is the complement of E i.e. that event E will not happen, then

P(E^{C}) = 1 – P(E)

**DeMorgan’s Law **

**Ex.**

let E_{1}, E_{2} are two events,

then

is the event either E_{1 } or E_{2 }or both.

is the event neither E_{1 }nor E_{2.}

De-Morgan’s law is often used to find the probability of neither E_{1 }nor E_{2.}

** **

**Corollary:1**

From theorem 3

If E^{C} is the complement of E, then

P(E^{C}) = 1 – P(E)

And from De-Morgen’s theorem

**Theorem 4:** If E = E_{1} ∪ E_{2} U E_{3} …. ∪ E_{n}, where E_{1}, E_{2}, ….. E_{n} are mutually exclusive events, then

P(E) = P(E_{1}) + P(E_{2}) + …… + P(E_{n}) = 1

In particular, if E = S, the sample space, then

P(E_{1}) + P(E_{2}) + ….. + P(E_{n}) = 1

**Theorem 5:** If A and B are any two events, then

P(E_{1} ∪ E_{2}) = P(E_{1}) + P(E_{2}) – P(E_{1} ∩ E_{2})

If both the events are Mutually Exclusive,

Then,

P(E_{1} ∩ E_{2}) = 0

Thus,

P(E_{1} ∪ E_{2}) = P(E_{1}) + P(E_{2})

More generally,

if E_{1}, E_{2}, E_{3} are any three events, then

P(E_{1} ∪ E_{2} ∪ E_{3}) = P(E_{1}) + P(E_{2}) + P(E_{3}) – P(E_{1} ∩ E_{2}) – P(E_{2} ∩ E_{3}) – P(E_{3} ∩ E_{1}) + P(E_{1} ∩ E_{2} ∩ E_{3})

**Theorem 6:** For any events E_{1} and E_{2},

P(E_{1}) = P(E_{1} ∩ E_{2}) + P(E_{1} ∩ E_{2}^{C})

Where B^{C} is compliment of B

**Theorem 7: ** If E_{1 }& E_{2} are two independent events, then

P(E_{1} ∩ E_{2}) = P(E_{1}) P(E_{2})

Then, P(E_{1} ∪ E_{2}) = P(E_{1}) + P(E_{2}) – P(E_{1} ∩ E_{2})

Will converts into P(E_{1} ∪ E_{2}) = P(E_{1}) + P(E_{2}) – P(E_{1})P(E_{2}) (for independent events)

**Theorem 8:** If an event E must result in the occurrence of one of the mutually exclusive events E_{1}, E_{2}, ….. E_{n}, then

P(E) = P(E ∩ E_{1}) + P(E ∩ E_{2}) + …. + P(E ∩ E_{n})

This is also known as Rule of total proabality

**Theorem 9: ****Conditional Probability**

Let E_{1} and E_{2} be two events such that P(E_{1}) > 0.

The probability of E_{2} given that E_{1} has occurred denoted by P(E_{2}/E_{1}) and given by,

This rule is also known as multiplication rule of probability.

- if E
_{1}& E_{2}are independent events

Then, P(E_{1} ∩ E_{2}) = P(E_{1}) P(E_{2})

For any three events E_{1}, E_{2}, E_{3},

we have

P(E_{1} ∩ E_{2} ∩ E_{3}) = P(E_{1}) (E_{2}| E_{1}) P(E_{3}| E_{1} ∩ E_{2})

In words, the probability that E_{1} and E_{2} and E_{3} all occur is equal to the probability that E_{1} occurs times the probability that E_{2} occurs given that E_{1} has occurred times the probability that E_{3} occurs given that both E_{1} and E_{2} have occurred.

**Example – 3 **In a certain college 25% of students failed in mathematics, 15% failed in chemistry & 10% failed in mathematics and chemistry both. If a student is selected at random

(i) If he failed in chemistry then what is the probability that he failed in mathematics too?

(ii) If he failed in mathematics then what is the probability that he failed in Chemistry too?

(iii) What is the probability that he neither failed in mathematics nor in chemistry.

**Sol.**

students failed in Mathematics = 25%, P(M) = 0.25

students failed in Chemistry = 15%, P(C) = 0.15

Students failed in mathematics and chemistry both = 10%, P(M ∩ C) =0.1

(i) If he failed in chemistry then what is the probability that he failed in mathematics too

**Example -4 **A box A contains 2 white and 4 black balls. Another box B contains 5 white and 7 black balls. A ball is transferred from the box A to the box B. Then a ball is drawn from the box B. Find the probability that it is white.

**Sol.**

The probability of drawing a white ball from box B will depend on whether the transferred ball is black or white.

If black ball is transferred from box A to box B, its probability is 4/6(probability of transferring the black ball). There are now 5 white and 8 black balls in the box B.

Then the probability of drawing white ball from box B is 5/13.

Thus, the probability of drawing a white ball from urn B, if the transferred ball is black

**Theorem 9: Baye’s Theorem**

It is an extended form of Conditional probability.

Suppose that E_{1}, E_{2}, E_{3} …….E_{m} are the mutually exclusive events whose union is the sample space and E is an event

Then, as per the baye’s theorem

In general form,

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