Probability - Session 5 and Session 6

In today's session, we started with the revision of previous classes. We also got to know about our exam pattern. We then started to discuss about Venn diagrams and probabilities.

There are 3 methods of estimation. They are:

• Ratio Chance : A ratio represents, for every amount of one thing, how much there is of another thing.Example 4/5
• Percentage Chance : In percentage, we talk about "x" out of 100. Percentage is reported as a ratio but the sum adds up to 100. Example 30% (30 out of 100)
• Probability : Is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen).The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen.  Example: Tossing a coin. The probability of head occurring is 0.5.  

History of Probability :The XVII-th century records the first documented evidence of the use of probability.More precisely in 1654 Antoine Gombaud, Chevalier deM´er´e, a French nobleman with an interest in gaming and gambling questions,was puzzled by an apparent contradiction concerning a popular dice game.

There are 3 approaches to calculate probability. They are:

• A priori : A priori probability can most easily be described as making a conclusion based upon deductive reasoning rather than research or calculation.

          Example : Consider how the price of a share can change. Its price can increase, decrease or remain               the same. Therefore, according to a priori probability, we can assume that there is a 1-in-3, or 33%,             chance of one of the outcomes occurring (all else remaining equal).

• Empirical Probability : Is one where we don't have prior information, but information is collected. Empirical probability is determined analytically, that is, by using our knowledge about the nature of experiment rather than through actual experiment. Empirical probabilities are point estimates and occur under one condition.

      Example : If you flip a coin ten times and get seven heads, your empirical probability is 7 in 10. 

                     If we assembled a return distribution based on the past 20 years of data, and then used that                          same distribution to make forecasts, we have used an empirical approach.

• Subjective Probability : A probability derived from an individual's personal judgment about whether a specific outcome is likely to occur. Subjective probabilities contain no formal calculations and only reflect the subject's opinions and past experience. Subjective probabilities occur under different conditions and have a range of values.

          Example :  Probability of getting a tender.

          We also worked out on a few problems based on the above mentioned approaches.

Sample Space

                   

                        A sample space is all possible outcomes of an event or an experiment.

       Example: A firm puts in tenders for 3 projects A, B & C. What is the sample space with respect to                              projects for which tenders are accepted.

       Sample Space, S={A, B, C, AB, AC, BC, ABC, Null}

Venn Diagram

             It is a diagram that shows all possible logical relations between a finite collection of sets. Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, statistics, logic etc.

Event Properties : There are 3 event properties. They are:

• Mutually Exclusive Events : Two events are mutually exclusive if they cannot occur at the same time. An example is tossing a coin once, which can result in either heads or tails, but not both. One more example of mutually exclusive events is profit and loss of a company.
• Independent Events : Two events are said to be independent if the outcome of one event doesn't impact the outcome of another. Two events A and B are independent if and only if their joint probability equals the product of their probabilities.Example of independent event: landing on head after tossing a coin and rolling a 5 on a single 6 sided dice.
• Dependent Events : Two events A and B are said to be dependent if the outcome of one event impacts the outcome of another event.Example for dependent events : From a pack of cards, the probability of picking a queen is 4/52 and the probability of picking a second card which is jack is 4/51. Here the outcome of the first event has impacted the outcome of the second event.

Laws of Probability : When we require the probability of two events occurring simultaneously or the probability of one or the other or both of two events occurring then we need probability laws to carry out the

calculations.

• Additional Law : 

  When two events, A and B, are mutually exclusive, the probability that A or B will occur is   the sum of the probability of each event.   P(A or B) = P(A) + P(B)   When two events, A and B, are non-mutually exclusive, the probability that A or B will         occur is: P(A or B) = P(A) + P(B) - P(A and B).

• Conditional Law : The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B|A). The formula for conditional probability is:

• Multiplicative Law : When two events, A and B, are independent, the probability of both occurring is P(A and B) = P(A) . P(B) .

     Bayes theorem

It is a direct application of conditional probabilities.  It is usually used to find the cause of a given experimental results.  P(Ai|B)=P(B|Ai)P(Ai)/∑j=ik(B|Aj)P(Aj). It was very exciting to learn a chapter that we were familiar with. We will continue to learn about probability in the forthcoming classes. References: Wikipedia Written By: Neeraj Ramadoss (2013167) Group Members: Nishanth Agarwal                          Nitin Shukla                          Prakar Swami                          Praveen Iyer                          Prerana Arora                          Neeraj Ramadoss