Probability

Definition:

An experiment is any well-defined operation or procedure that results in one of two or more possible outcomes. An outcome is particular result of an experiment.

Counting Techniques:

(a) Tree Diagram,

(b) Multiplication Rule,

(d) Combination.

(a) Tree Diagram: Counting the number of possible outcomes of an experiment plays a major role in probability theory. These possible outcomes can be shown by the branches of a tree-like diagram called ‘Tree-Diagram’ or ‘Branch Diagram’.

(b) Multiplication Rule: If an operation of an experiment can be performed in n₁ ways and if for each of these ways another operation can be performed in n₂ ways, then the two operations can be performed together in n₁ × n₂ ways, and the k^th in n_k ways, then all the k operations can be performed together in n₁ × n₂ × n₃ × …….. × n_k ways.

For example, a coin and a die tossed together, the possible outcomes will be:

n₁ = 2 (coin: two sides)

n₂ = 6 (die: 6 sides)

The two operations can result in (n₁ × n₂) 12 ways.

(c) Permutations: A permutation is a group of items with a certain ordered arrangement. For example: ABC, ACB, CAB, CBA and BCA are different permutations. The rules of permutation are different under each of the following four situations:

Situation I:

(i) All the items are distinct,

(ii) Each item can occur only once in an arrangement, i.e., repetition not allowed or without replacement, and

(iii) Each item can occupy any place in an arrangement.

In the above situation, the number of permutations of n items arranged r at a time, denoted by is:

Where r ≤ n

Example:

How many three-digit numbers can be formed from the digits 1, 2, 4, 5 and 9 without replacement?

Solution:

n = 5 and r = 3

Situation 2:

(i) All the items are distinct,

(ii) An item can be repeated in an arrangement (i.e., repletion allowed or with replacement), and

(iii) Each item can occupy any place in an arrangement.

In the above situation, the number of permutations of n items arranged r at a time is:

Example:

Arrange the license plate with 3 alphabets and 3 digits with replacement.

Solution:

Situation 3:

For n non-distinct items out of which n₁ are of one kind, n₂ are of another kind, ……, n_k are of another, and n₁ + n₂ + …………… + n_k = n, the number of permutations of all n items is:

Example:

Find the possible permutations of 7558.

Solution:

n = 4 (7558),

n₁ = 1 (one 7),

n₂ = 2 (two 5), and

n₃ = 1 (one 8).

Situation 4:

When the items are arranged in a circle, two arrangements are not considered different, unless corresponding items of both are preceded or followed by a different item.

The number of permutations of n distinct items arranged in a circle is:

= (n – 1)!

The above lines have different permutations:

The above circles have same permutations.

Example:

Arrange 5 different trees in a circle:

Solution:

n = 5 trees

= (n – 1)!

= (5 – 1)! = 4! = 24 ways

(d) Combinations: A combination is a group of items without regard to the arrangement of items. ABC and BCA are two different permutations but are same combinations of three letters.

The number of combination of n distinct items taken r at a time, denoted by , is:

Example:

In how many ways five students can be selected from a group of 12 students?

Solution:

n = 12; r = 5

Basic Concepts of Probability Theory:

(a) Possibility Space,

(b) Event,

(d) Mutually Exclusive Events,

(e) Composite Events, and

(f) Joint Events.

(a) Possibility Space: Also known as ‘sample space’ or ‘outcome space’. A possibility space is a set of all possible outcomes of an experiment and is denoted by S.

If an experiment consists of a toss of a fair die and the numbers are of interest, the possibility space would be:

S = {1, 2, 3, 4, 5, 6}

If the interest is whether the number is even or odd, the possibility space would be:

S = {even, odd}

A possibility space may be represented by a rectangle. The number of outcomes in the possibility space is denoted by n(S).

(b) Event: A subset of a possibility space is called an event and is usually denoted by first few capital letters A, B, C, ……… For example, a coin is tossed, the sample space is S = {H, T} and the subset A = {H} is the event when a head occurs.

Take another example, two coins are tossed. The sample space is S = {HH, HT, TH, TT} and the subset B = {HH, HT, TH} is the event that at least one head appears when two coins are tossed.

An event may be represented by a circle inside the rectangle of the possibility space.

An event is further divided into the following:

(i) Simple Event: is the subset if it contains only one outcome of the possibility space.

(ii) Compound Event: is the subset if it contains more than one outcomes of the possibility space.

(iii) Null Event: is a subset containing no outcomes. It is also called ‘impossible event’.

(iv) Sure Event: It is also called ‘certain event’. It is a subset containing all outcomes of the possibility space.

Number of possible events = 2ⁿ

Example:

Two coins are tossed. List all the possible events or subsets of the possibility space.

Solution:

S = {HH, HT, TH, TT}

n(S) = 4

Number of possible events = 2ⁿ = 2⁴ = 16 events

Possible Events of two tossed coins:

(c) Complementary Event: For an event A, the complementary event is defined as a set of those outcomes of the possibility space which are not in A. The complementary event of A is written as (not A).

For example, the complementary events of E₅ = {TT} are . It is diagrammatically shown as below:

(d) Mutually Exclusive Events: Two events A and B are mutually exclusive if they have no outcomes in common and therefore they cannot happen together. Mutually exclusive events are also known as disjoint events.

Non-mutually exclusive events are the vice versa of the above definition and are also known as over-lapping events.

For example, in case of two tossed coins, E₆ and E₇ are not mutually exclusive events, because both events have an outcome HH in common. However, E₆ and E₁₁ are two mutually exclusive events because no single outcome is common.

(e) Composite Events: For two events A and B, a composite event is defined as a set of outcome of either A or B or both A and B and therefore at least one of two events must occur. The composite event of A and B is written as AUB, or A or B.

For example, the composite event of E₆ and E₇ is as follows:

E₆ U E₇ = {HH, HT, TH}

(f) Joint Events: For two events, A and B, a joint event is defined as a set of common outcomes of A and B and therefore both the events must occur together. The joint event of A and B is written as ‘A and B’ or ‘A∩B’.

For example, the joint event of E₆ and E₇ is as follows:

E₆ ∩ E₇ = {HH}

Take another example, the joint event of E₂ and E₁₅ is as follows:

E₂ ∩ E₁₅ = { } (i.e., Null Event)

Probability Theory:

1. A probability is a numerical measure of the likelihood (or chance) that a particular event will occur.

2. The probability of any event must satisfy the following two conditions.

(i) No probability is negative, P(Event) ≥ 0.

(ii) No probability is greater than one, P(Event) ≤ 1.

3. There are three different approaches to assign probabilities to the events:

(a) Classical or Mathematical Approach

(b) Empirical or Relative Frequency Approach

(a) Classical Approach: It is the approach in which probabilities are assigned to the events before the experiment is actually performed and therefore, such probabilities are also called ‘a priori’ probabilities.

If the possibility space of the experiment is finite, and if each outcome of the possibility space is equally likely to occur, then the probability of event A:

It is also referred to as ‘axiomatic function of probability’.

Example:

Two coins are tossed once, what is the probability that two heads will appear?

Solution:

S = {HH, HT, TH, TT}

n(S) = 4

Event = Two head appear

i.e., n(A) = 1

Now the probability of event A is as calculated below:

(b) Relative Frequency Approach: This approach is applied when the possibility spaces are infinite, or the outcomes cannot be assumed equally likely.

If an experiment is represented ‘n’ times under uniform conditions and if ‘m’ times the outcome of the experiment is in favour of an event A, then the ratio approaches the probability of the event A as ‘n’ approaches infinity.

The ratio is considered as an estimate of the actual probability of event A and normally this estimate is called the probability of the event A and is written as:

Since in this approach probabilities are assigned after performing a large experiment, therefore, they are also known as ‘a posteriori’ probabilities.

Example:

A die has been rolled 360 times and ‘Six’ has been observed 63 times. Estimate the probability of occurring a ‘Six’ when the die is to be rolled once again.

Solution:

m = 63

n = 360

(c) Subjective Approach of Probability: Subjective probability can be defined as the probability assigned to an event by an individual, based on whatsoever evidence is available. This evidence may be in the form of relative frequency of past occurrences, or it may be just an educated guess.

Probability of Complementary Events:

If A and are complementary events in a probability S, then:

Example:

A coin is tossed 3 times. Find the probability of getting at least one tail.

Solution:

Number of possible outcomes in S = n(S) = 2³ = 8

Let A = At least one tail appears

Then = No tail appears = All heads appear

(the only outcome with all heads)

Independent and Dependent Events:

1. When two events are given, the occurrence of the first event may or may not have an effect on the occurrence of the second event.

2. When the occurrence of one of the two events has no effect on the probability of the occurrence of the other event, the two events are called ‘Independent Events’.

3. On the other hand, when the occurrence of first event has some effect on the probability of occurrence of second event, the second event is said to be ‘Dependant’ on the first event.

Conditional Probability:

If there are two events A and B such that the probability of event B depends on the occurrence or non-occurrence of the event A, then the probability of event B occurs when the event A occurs is called the ‘conditions probability of event B given event A’ and is written as:

and

If two events are independent then:

P(B/A) = P(B) (independent events)

Example:

A black card is drawn from an ordinary deck of 52 playing cards. What is the probability that it is of spade (♠) suit?

Solution:

Let B = Black card drawn

and S = Spade card drawn

Since the card drawn is black (event B has occurred), and there are 13 spade cards in 26 black cards, therefore:

Multiplication Law of Probability:

If there are two non-mutually exclusive events A and B such that event B is dependent on event A, then the probability of joint event (A and B) is given by:

P(A and B) = P(A) × P(B/A)

P(A∩B) = P(A) × P(B/A) or P(A) + P(B) – P(AUB)

or it may equivalently be written as:

P(A∩B) = P(B) × P(A/B)

When two events A and B are independent:

P(A∩B) = P(A) × P(B)

When two events A and B are mutually exclusive, their joint probability is zero:

P(A∩B) = 0

Example:

In a graduate college, there are 500 male and female students learning B.Sc and B.Com. The break up is as follows:

	Male	Female	Total
B.Sc	70	150	220
B.Com	120	160	280
Total	190	310	500

What is the probability that a randomly selected student is (i) a female B.Sc student, and (ii) a male B.Com student?

Solution:

(i) a female B.Sc student:

Let A = student selected is learning B.Sc;

B = student selected is a female;

A∩B = female student learning B.Sc

P(A∩B) = P(A) × P(B/A), where

(ii) a male B.Com student:

Let = student selected is not learning B.Sc;

= student selected is not a female;

Addition Law of Probability:

If A and B are two non-mutually exclusive events, then the probability of the composite event (A or B) is given by:

P(A or B) = P(A) + P(B) – P(A and B)

P(AUB) = P(A) + P(B) – P(A∩B)

For two mutually exclusive events A and B:

P(AUB) = P(A) + P(B)

Example:

A card is drawn from a pack of 52 cards. What is the probability that:

(i) it is either Ace or King

(ii) it is either Queen or a Diamond (♦)

Solution:

(i) it is either Ace or King:

(AUB) = the card drawn is either an Ace or a King

PAUB) = P(A) + P(B)

(ii) it is either Queen or a Diamond:

CUD = Card drawn is either a Queen or a Diamond

P(CUD) = P(C) + P(D) – P(C∩D); where P(C∩D) = P(C) × P(D/C) =

P(CUD)

Baye’s Theorem:

The Baye’s Theorem is based on conditions probabilities. It calculates probabilities of the causes that may have produced an observed event.
Given A₁, A₂, ………, A_i, …………., A_n mutually exclusive events, whose union is the entire possibility space S, and let B an arbitrary event in S, such that P(B) ≠ O, then:

Where i = 1, 2, 3, 4, ……., n

P(A_i) are called ‘prior probabilities’ and P(A_i/B) are called ‘posterior probabilities’. Thus Baye’s Theorem is a process to revise the prior probabilities using additional sample information and get the posterior probabilities.

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