PROBABILITY 1

Basic Terminology 2

Probability 3  Probability is the numerical measure of the likelihood that an event will occur  The probability of any event must be between 0 and 1, inclusively Certain Impossible ≤ P(A) ≤ 1 For any event A

Random Experiment 4  If an experiment is conducted repeatedly under essential homogeneous conditions, the results are not unique but may be one of the various possible outcomes.

Trial 5  Performing of a random experiment is called a trial.

Sample Space 6 The Sample Space is the collection of all possible events e.g. All 6 faces of a die: e.g. All 52 cards in a pack of playing cards.

Event 7  It is defined as the outcomes or combination of outcomes of an trial in an random experiment.

Simple event 8  Events are defined as simple if it corresponds to a single outcome in an experiment.

Compound Event 9  Events are defined as compound if they correspond to at least two or more outcomes of the experiment.

Mutually Exclusive Events 10 If the two events A and B are such that they cannot happen together in an experiment, then they are said to be Mutually exclusive. If the two events A and B are such that they cannot happen together in an experiment, then they are said to be Mutually exclusive.

Mutually Exclusive Events 11

Completely Exhaustive Events 12 Events are called exhaustive if their union completely covers the all possible outcomes in the sample space. Events are called exhaustive if their union completely covers the all possible outcomes in the sample space.

Definition of Probability 13  The various definitions are :- (i)Classical (ii) Statistical or Empirical (iv) Subjective

Classical Definition 14  If an event can happen in m ways out of a total of n possible equally likely ways, the probability of its happening is defined as m P ( E ) = P ( E ) = n

Statistical or Empirical Definition 15  Empirical probability of an event is defined as the relative frequency of occurrence of the event when the number of observations is very large.

Subjective Probability 16  Subjective probability is the subjective estimates of probabilities of some event. It differs from person to person based on experience, perceptions, and above all judgmental capabilities. It is used when it is not possible to gather requisite data. It is used when it is not possible to gather requisite data.

Union & Intersection 17 Following notations are used universally. (A U B): Read as A union B (A U B): Read as A union B and implies Either A or B or both A and B

Union & Intersection 18 ( A ∩ B ): Read as A intersection B and implies Both A and B and implies Both A and B

Independent Events 19  Two Events are said to be Statistically independent if the happening or non- happening of one does not affect, and is not affected by the happening or non-happening of the other.

Dependent Events 20  Two Events are said to be Statistically dependent if the happening or non-happening of one does affect, and is affected by the happening or non-happening of the other.

Probability under Statistical Interdependence 21  Marginal probability: It is simply the probability of occurrence of that event. e.g. Tossing of a coin; e.g. Tossing of a coin; P(H) = 0.5, and P(T) = 0.5 P(H) = 0.5, and P(T) = 0.5

Probability under Statistical Interdependence 22 Joint probability: It is the probability of occurrence of two or more events simultaneously. In independence case it is the product of their respective marginal probabilities. P(AB) = P(A) x P(B) e.g. Getting Heads in two toss of a coin. P(HH) = ½ * ½ = ¼

Probability under Statistical Interdependence 23 Conditional probability: As per definition, conditional probability of event A, P(A/B) is defined as the probability of happening of A provided B has already occurred. In independence case, In independence case, P(A/B) = P(A) P(A/B) = P(A)

Probability under Statistical Dependence 24 Marginal probability: It is computed as the sum of all the joint events in which simple event occurs. E.g. Suppose a bag contains10 balls such as, Three are colored and dotted One is colored and stripped Two are grey and dotted Four are gray and stripped.

Probability under Statistical Dependence 25 Now probability of colored ball P(C) is given as, P(C) = P(CD) + P(CS) P(C) = P(CD) + P(CS) = = 0.4 = = 0.4

Probability under Statistical Dependence 26 Joint probability: For two events A & B, as per the definition of conditional probability, it is given as, P(AB) = P(A/B) x P(B) P(AB) = P(A/B) x P(B) or = P(B/A) x P(A) or = P(B/A) x P(A)

Probability under Statistical Dependence 27 Joint probability In the previous example it can be given as probability of drawing a ball which is colored and stripped. E.g. P(CS) = P(C/S) * P(S) = 0.2 * 0.5 = 0.1 = 0.2 * 0.5 = 0.1

Probability under Statistical Dependence 28 Conditional probability: For two events A & B, it is given as, P(A/B) = P(AB) / P(B)

Probability under Statistical Dependence 29 Conditional probability In the previous example it can be given as probability of colored ball provided the drawn ball is stripped. In the previous example it can be given as probability of colored ball provided the drawn ball is stripped. E.g. P(C/S) = P(CS) / P(S) = 0.1/ 0.5 = 0.2 = 0.1/ 0.5 = 0.2

Addition Theorem of Probability 30 P(AUB) = P(A) + P(B) - P(A ∩ B)

Multiplication Theorem of Probability 31 P(A ∩ B) = P (A) P (B / A) P(A ∩ B) = P (A) P (B / A) or, or, P(AB ) = P(A) P (B) if A and B are independent events.

Baye’s Theorem – A Priori and Posterior Probabilities 32 Now, suppose, it is known that the event A has happened but it is not known whether it is due to B 1,B 2,B 3, …., Bn, i.e. which of the Bi was the cause for the happening of A. Bayes’ Theorem gives this information in terms of probabilities as follows : Now, suppose, it is known that the event A has happened but it is not known whether it is due to B 1,B 2,B 3, …., Bn, i.e. which of the Bi was the cause for the happening of A. Bayes’ Theorem gives this information in terms of probabilities as follows :

Baye’s Theorem 33 P ( Bi / A ) ( i = 1, 2, …, n) Here, P(Bi)s are called a priori probabilities, and P(Bi/A)s are called posterior probabilities, that indicate the change in a priori probabilities because of the additional information that A has occurred. )/()( )/()( ii ii BAPB P BAPBP  