Probability

Probability

DEFINITIONS

Random Experiment

Random Experiment is an experiment whose all possible outcomes (results) are known in advance, but the result of any specific performance cannot be predicted

before completion of the experiment.

Sample Space

The set of all possible outcomes of a random experiment is said to be the sample space of the experiment. It is usually denoted by S. Then the sample space S of

the experiment is the finite set S {e₁, e₂, ...., e_m}. Each element of the set is a possible outcome and each outcome of a trial corresponds to only one element

of the set S.


Event

Any subset of the sample space is called an event. Any event associated with a random experiment can be represented by a non-empty subset E of S. The event

E is said to have occurred if the outcome e of the experiment satisfies e E.


Equally likely events/outcomes

In a random experiment, two or more events/ outcomes are said to be equally likely, if no one of them has more chance of occurrence than others. In other words

each event of S has same chance of occurring.


Mutually Exclusive events

In an experiment, two or more events are said to be mutually exclusive, if the occurring of one of them rules out the possibility of the occurring of others. i.e. they

cannot occur simultaneously


Exhaustive set of events

A set of outcomes / events is said to be exhaustive if it covers each and every possible outcome of the sample space of the experiment. i.e. at least one of them

has to occur in a performance of the experiment. If A₁, A₂, .... , A_n are events defined on a sample space S, and A₁ A₂A₃ ... A_n = S, then

these events are called as exhaustive events ( i.e one of these must occur).

An empty set is always a subset of a set S. Hence, the empty set can be considered as representing an event of an experiment. But there is no

experiment without an outcome. , thus represents an impossible event.


We define the event "not A", as the event which occurs when A does not occur. Often 'not A' is

called the complementary event of A, or negation of A. Since A A^c = , event A and its complementary A^c are mutually exclusive.



PROBABILITY

Associated with each outcome in a sample space is a real number p(0 p 1) called the probability of that outcome which represents the chance of

happening of the event.


Definition:

An event is said to have an associated probability p defined by

p = .

Therefore if an event can happen in 'a' ways and fail in 'b' ways (the total no. being a + b) and if all the ways are equally likely, then the probability of happening of

the event = and the probability of non-occurrence of the event = .

(Note that: p + q = 1, Also 0 p, q 1). The probability of an event A is denoted by P (A).


Rule 1:

In the case of mutually exclusive events the chance of happening of one or other of them is the sum of the chances of the separate events. e.g. if A and B are

mutually exclusive events, P(AB) = P(A) + P(B).


Conditional Probability

Rule 2:

P(A/B) denotes the probability of event A happening, given that event B has happened. It is the conditional probability of A, given B. While calculating P(A/B),

we assume that event B has occurred. It implies that the outcomes favourable to B become the total outcomes and hence outcomes favourable to P(A/B) are

outcomes common to A and B.

Let n be the number of events in the sample space. Let m₁ denote the number of elementary events favourable to B which are also favourable to A and m₂ denote

the number of elementary events favourable to B only. Obviously m₂ m₁ and.

P(A/B) = P(A given B)

= .

Thus P(A/B) =, P (B) 0.

Similarly

(P(AB) = P(B). (P(A/B) = P(A). (P(B/A)).

Note that if A and B are independent events then

P(AB) = P(A) P(B)

P(A/B) =

which should be the case as occurrence of A does not depend upon B.

Similarly P(B/A) = P(B).


Independent Events

Events are said to be independent when the happening of any one of them does not affect the happening of any of the others.

(i).A and B are independent if

P(B/A) = P(B) and P(A/B) = P(A).

P(A and B) = P(A B) = .


(ii).The probability of the concurrence of several independent events is the product of their individual probabilities.

i.e. if E₁, E₂,...,E_n are n independent events then P(EE₂...E_n)

= P(E₁)P(E₂)P(E₃)...P(E_n).


Remark:

The converse of this is also true i.e. if n given events satisfy the above condition then they will be independent.
If the events A and B are independent, then and are also independent.



Pairwise Independent Events

Three events E₁, E₂ and E₃ are said to pairwise independent if

P(E₁ E₂) = P(E₁) P(E₂), P(E₂ E₃) = P(E₂) P(E₃) and P(E₃ E₁) = P(E₃) P(E₁)

i.e. Events E₁, E₂, E₃, ...., E_n will be pairwise independent if

P(E_i E_j )= P(E_i) P(E_j) i j.

Three events are said to be mutually independent if

P(E₁E₂) = P(E₁)P(E₂), P(E₂E₃) = P(E₂) P(E₃) and P(E₃E₁) = P(E₃) P(E₁)

and P(E₁E₂E₃) = P(E₁) P(E₂) P(E₃).

If two events A and B are mutually exclusive,

P(AB) = 0 but P(A) P(B) 0 ( In general).

P(AB) P(A) P(B)

mutually exclusive events will not be independent.

We conclude that if two events are independent, they have to have some common element between them i.e they cannot be mutually exclusive. Mutually

exclusiveness is used when the events are taken from the same experiment and independence is used when the events are taken from different experiments.


Total Probability Theorem

Let A₁, A₂, ..., A_n be a the set of mutually exclusive and exhaustive events and E be some event which is associated with A₁, A₂, ..., A_n. Then probability that E

occurs is given byP(E) = .



Baye's Theorem

Suppose A₁, A₂, ... , A_n are mutually exclusive and exhaustive set of events. Thus, they divide the sample space into n parts and an event B occurs.

Then the conditional probability that A₁ happens (given that B has happened) is given by P(A₁/B) = .


Note: In conditional probability the sample space is reduced to the set of samples/ outcomes in the event which is given to have happened.


Probability distribution of a random variable

Consider the random experiment of tossing two coins. The sample space of the outcome is

S {HH, HT, TH, TT}.

After completing the experiment, we count the number of tails that have turned up. Let this number be X.

If X = 0, no tails has turned up i.e. H, H have occurred.

If X = 1, onee tail has turned up i.e. either H, T or T, H have occurred.

If X = 2, two tails have turned up i.e. T, T have occurred.

We write these details as



We note that a numerical value 0, 1 or 2 is assigned to each sample point of the sample space. Thus, numbers 0, 1, 2 are the values of the variable X denoting number of tails in each of the outcome of the experiment. We call this variable a random variable.

A random variable is thus a variable whose values are numbers determined by the outcomes of a random experiment. Let us now write the probability of each outcome in the above example. We have

P (HH) =, P (HT) =, P (TH) =, P (TT) =

P (X = 0) =, P (X = 1) = + =, P (X = 2) =.

The system consisting of a random variable X, along with P (X) is called the probability distribution of X. Let the random variable X take values x₁, x₂, ....., x_n with probabilities

P (X = x₁) = p₁, P (X = x₂) = p₂ ..... P (X = x_m) = p_m.

The probability distribution is

X x₁ x₂ x₃ ........ x_m

P (X) p₁ p₂ p₃ ........p_m
The function P(X) is called the probability function X. Here p_i > 0, i = 1, 2, .... , m and .


Binomial Distribution for Successive Events

Suppose p and q are the respective chances of the happening and failing of an event at a single trial (q = 1 - p). Then the chance of its happening r times (exactly) in n trials is ⁿC_r p^rq^n-r because the chance of its happening r times and failing n - r times in a given order is p^rq^{n - r} and there are ⁿC_r such orders, which aremutually exclusive and for any such order, the probability is p^rq^{n - r}. Therefore, the required probability is ⁿC_r p^rq^n-r.

Note that the probabilities P(x) are given by the terms in the binomial expansion of (p + q)ⁿ . Because of this, the probability distribution is called as a binomial distribution.