We like to think that we have control over our lives. But in reality there are many things that are outside our control. Everyday we are confronted by our own ignorance. The world is governed by Quantum Mechanics where Probability reigns supreme. Why study Probability?

You wake up in the morning and the sunlight hits your eyes. Then suddenly without warning the world becomes an uncertain place.  How long will you have to wait for the Number 13 Bus this morning?  When it arrives will it be full?  Will it be out of service?  Will it be raining while you wait?  Will you be late for your 9am Maths lecture? Consider a day in the life of an average student.

 It is used by Physicists to predict the behaviour of elementary particles.  It is used by engineers to build computers.  It is used by economists to predict the behaviour of the economy.  It is used by stockbrokers to make money on the stockmarket.  It is used by psychologists to determine if you should get that job. Probability is the Science of Uncertainty.

 Can you make money playing the Lottery?  Let us calculate chances of winning.  To do this we need to learn some basic rules about probability.  These rules are mainly just ways of formalising basic common sense.  Example: What are the chances that you get a HEAD when you toss a coin?  Example: What are the chances you get a combined total of 7 when you roll two dice? Consider a Real Problem

An Experiment leads to a single outcome which cannot be predicted with certainty. Examples- Toss a coin: head or tail Roll a die: 1, 2, 3, 4, 5, 6 Take medicine: worse, same, better Set of all outcomes - Sample Space. Toss a coin Sample space = {H,T} Roll a die Sample space = {1, 2, 3, 4, 5, 6} Experiments

EVENT: An outcome or a set of outcomes of an activity Example:In tossing a coin, getting a head COMPOUND EVENT: The joint occurrence of two or more simple events Example:If a bag contains 3 Red and 3 Blue balls and two balls are drawn randomly,then event of pulling out 1 red and 1 blue ball is a compound event. SAMPLE SPACE: Collection of all possible events or outcomes of an experiment

The Probability of an outcome is a number between 0 and 1 that measures the likelihood that the outcome will occur when the experiment is performed. (0=impossible, 1=certain). Probabilities of all sample points must sum to 1. Long run relative frequency interpretation. EXAMPLE: Coin tossing experiment P(H)=0.5 P(T) = 0.5 Probability

Example: THE GAME Of CRAPS In Craps one rolls two fair dice. What is the probability of the sum of the two dice showing 7?

Sample space S = (1,1)(1,2)(1,3)(1,4)(1,5)(1,6) (2,1)(2,2)(2,3)(2,4)(2,5)(2,6) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) (6,1)(6,2)(6,3)(6,4)(6,5)(6,6) So the Probability of getting the sum as 7 when rolling two dice is {(1,6), (2,5), (3,4), (4,3),(5,2), (6,1) } = 6/36 = 1/6

Equally likely outcomes In a Sample Space S of equally likely outcomes. The probability of the event A is given by P(A) = n(A )/n(S) That is the number of outcomes in A divided by the total number of events in S.

A compound event is a composition of two or more other events. A C : The Complement of A is the event that A does not occur A  B : The Union of two events A and B is the event that occurs if either A or B or both occur, it consists of all sample points that belong to A or B or both. A  B: The Intersection of two events A and B is the event that occurs if both A and B occur, it consists of all sample points that belong to both A and B Sets

Basic Probability Rules P(A c )=1-P(A) P (A or B)=P(A  B)=P(A)+P(B)-P(A  B) Mutually Exclusive Events are events which cannot occur at the same time.  In an unbiased coin tossing experiment, head and tail cannot occur together.  When rolling a dice,3 and 4 cannot appear together P(A  B)=0 for Mutually Exclusive Events. P(A intersection B) = P(AB)=P(A and B)

Independent Events A and B are independent events if the occurrence of one event does not affect the occurrence of the other event. Example: P(prefer engineering education / Gender) = P(prefer engineering education Because the preference for an engineering education is independent of gender If A and B are independent then P(A|B)=P(A) P(B|A)=P(B) P(A  B)=P(A)P(B)

Formulae CONDITIONAL PROBABILITY P(A | B) ~ Probability of A occuring given that B has occurred P(A | B) = P(A  B) / P(B) BAYES’ THEOREM

PROBABILITY ASSIGNING TECHNIQUES 1.CLASSICAL TECHNIQUE -A mathematical approach of assigning probability -The probability of occurrence of an event P(E) = Number of favourable cases/ the exhaustive number of cases (total possible outcomes) Example:A company employs a total of 400 workers.Out of this, 150 are skilled and 250 are unskilled.The probability of selecting a skilled worker is 150/400 = 0.375

2.RELATIVE FREQUENCY TECHNIQUE -Uses the relative frequencies of past occurrences as the basis of computing present probability -Past data is used to predict future possibility -Probability is defined as the proportion of times an event occurs in a large number of trials

Example A company retains a team of 10 quality control inspectors for maintaining good quality of raw materials.As per past data, the team has rejected 10 batches out of 50.what is the probability that this team is going to reject the new batch of raw materials? P(E) = 10/50 = 0.2

3.SUBJECTIVE APPROACH - Based on the intuition of an individual -Based on the accumulation of knowledge, understanding, or experience of an individual -Not based on Mathematics Example: A sales manager wants to promote one of his subordinates out of three.All are equal interms of efficiency,punctuality,selling potential,behavioural aspects etc.,The sales manager selects the best person on his intuition.