Probability:

Probability concerning numerical description of how likely an event is to occur.

Types of Probability:

1.Classical Probability:

In this type of Probability the outcomes can be calculated by listing.

Ex.Tossing a coin

2.Experimental/Empirical Probability:

This is based on the number of possible outcomes by total number of trials.
Ex.The probability of a coin which is flipped 50times and landed on heads 28 times.

3.Subjective Probability:

It is based on the person's own personal reasoning and judgement.
Ex.A Cricket Fan may predict his/her favourite team score and guess that it will win.

4.Theoretical Probability:

It is an approach on the basis of the possible probability on possible chances of something happen.
Ex.The probability of getting 3 in a dice.

5.Axiomatic Probability:

Axiomatic probability is a unifying probability theory. It sets down a set of axioms (rules) that apply to all of types of probability, including frequentist probability and classical probability. These rules, based on Kolmogorov's Three Axioms, set starting points for mathematical probability. (Which is discussed below)

Elements of a Probabilistic Model:

1.Experiment:

Which work/things gives uncertain outcomes is called Experiment.
Ex.Tossing a coin

2.Sample Space:

The set of all possible outcomes of an experiment.

3.Event:

A event which is a subset of sample space and non negative number that assign the given outcomes or required outcomes.

4.P(A):

It is called probability of a that ensure the ratio of events to the sample space.
It will always less that 1.

Types Of Events:

1. Simple Event/Elementary Event

If the event E has only one sample point of a sample space, it is called a simple event or an Elementary Event. It is an event that consists of exactly one outcome. Let us understand this with an example. Say you throw a die, the possibility of 2 appearing on the die is a simple event and is given by E = {2}.

2. Compound Event

As opposed to a simple event, if there is more than one sample point on a sample space, such an event is called Compound Event. It involves combining two or more events together and finding the probability of such a combination of events.

For example, let us take another example. When we throw a die, the possibility of an even number appearing is a compound event, as there is more than one possibility, there are three possibilities i.e. E = {2,4,6}.

3. Certain Event/Sure Event

Just as the name suggests, an event which is sure to occur in any given experiment is a certain event. The probability of this type of event is 1.

4. Impossible Event

On the other hand, when an event cannot occur i.e. there is no chance of the event occurring it is said to be an impossible event. The probability of this event is 0. Like the probability that the card you drew from a deck is both red and black is an impossible event.

5. Equally likely Events

When the outcomes of an experiment are equally likely to happen, they are called equally likely events. Like during a coin toss you are equally likely to get heads or tails.

6.Complimentary Events

The non-occurence of an event is called complimentary event. It is denoted by " A' " or " A^c"

7. Mutually Exclusive Events

Two events A and B is called Mutually Exclusive events if A⋂B=ф

i.e; They have no common elements.

Probability Axioms:

(Non-negativity) P(A) ≥ 0 for every event A.
(Additivity) If A and B are two disjoint events,then the probability of their union is P(A U B)= P(A) + P(B)
(Normalization) The probability of entire Sample space ᘯ is equal to 1. i.e;P(ᘯ)=1

Conditional Probability

The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred. This probability is written P(B|A), notation for the probability of B given A. In the case where events A and B are independent (where event A has no effect on the probability of event B), the conditional probability of event B given event A is simply the probability of event B, that is P(B).

[Note: P(A) in P(A|B) is called Prior Probability]

Example:

Independence:

Two events

A

and

B

are independent if and only if

P (A \cap B) = P (A) P (B)

EXAMPLE:

Law of Total Probability

For two events A and B associated with a sample space S, the sample space can be divided into a set A ∩ B′, A ∩ B, A′ ∩ B, A′ ∩ B′. This set is said to be mutually disjoint or pairwise disjoint because any pair of sets in it is disjoint. Elements of this set are better known as a partition of sample space.

This can be represented by the Venn diagram as shown in fig. 1. In cases where the probability of occurrence of one event depends on the occurrence of other events, we use total probability theorem.

In Other Word

EXAMPLE:

I have three bags that each contain $100$ marbles:

Bag 1 has $75$ red and $25$ blue marbles;
Bag 2 has $60$ red and $40$ blue marbles;
Bag 3 has $45$ red and $55$ blue marbles.

I choose one of the bags at random and then pick a marble from the chosen bag, also at random. What is the probability that the chosen marble is red?

Solution:

Let $R$ be the event that the chosen marble is red. Let $B_{i}$ be the event that I choose Bag $i$ We already know that

P (R | B 1) = 0.75,

P (R | B 2) = 0.60,

P (R | B 3) = 0.45

We choose our partition as $B_{1}, B_{2}, B_{3}$ . Note that this is a valid partition because, firstly, the $B_{i}$ 's are disjoint (only one of them can happen), and secondly, because their union is the entire sample space as one the bags will be chosen for sure, i.e., $P (B_{1} \cup B_{2} \cup B_{3}) = 1$ Using the law of total probability, we can write

P (R)

= P (R | B_{1}) P (B_{1}) + P (R | B_{2}) P (B_{2}) + P (R | B_{3}) P (B_{3})

= (0.75) \frac{1}{3} + (0.60) \frac{1}{3} + (0.45) \frac{1}{3}

= 0.60

Bayes' Rule

suppose that we know $P (A | B)$ , but we are interested in the probability $P (B | A)$ . Using the definition of conditional probability, we have

P (A | B) P (B) = P (A \cap B) = P (B | A) P (A) .

Dividing by

P (A)

, we obtain

P (B | A) = P ( A | B ) P ( B )/ P ( A ),

which is the famous Bayes' rule. Often, in order to find

P (A)

in Bayes' formula we need to use the law of total probability, so sometimes Bayes' rule is stated as

P (B j | A) = P ( A | B j ) P ( B j )/ \sum i P ( A | B i ) P ( B i ),

where

B_{1}, B_{2}, \dots, B_{n}

form a partition of the sample space.

Example:
In the above example suppose we observe that the chosen marble is red. What is the probability that Bag 1 was chosen?

Conditional Independence

Two events

A

and

B

are conditionally independent given an event

C

with

P (C) > 0

P (A \cap B | C) = P (A | C). P (B | C)

In very Clear Language

The conditional Probability of A given B is represented by P(A|B). The variables A and B are said to be independent if P(A)= P(A|B) (or alternatively if P(A,B)=P(A) P(B) because of the formula for conditional Probability).

Example 1 Suppose Norman and Martin each toss separate coins. Let A represent the variable "Norman's toss outcome", and B represent the variable "Martin's toss outcome". Both A and B have two possible values (Heads and Tails). It would be uncontroversial to assume that A and B are independent. Evidence about B will not change our belief in A.

MatheMedium

Pages

Friday, June 19, 2020

Introduction to Probability (Chapter-1)