18th july lecture (PROBABILITY)
The PROBABILITY of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.
3 approaches to calculate probability are :-
1). A priori approach (prior information )
It is the Probability calculated by logically examining existing information. A priori probability can most easily be described as making a conclusion based upon deductive reasoning rather than research or calculation. The largest drawback to this method of defining probabilities is that it can only be applied to a finite set of events.
For 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).
One disadvantage of A priori probability is that it applies only to finite collections of events.
2). Empirical approach(Information collected)
A form of probability that is based on some event occurring, which is calculated using collected empirical evidence. An empirical probability is closely related to the relative frequency in a given probability distribution.
An example for empirical method would be the study of capital asset pricing model which is often used to estimate a companies weighted average cost of capital.
3). Subjective approach (judgemental)
Sample Space
The set of all possible outcomes of an experiment is called the sample space.
We will denote the outcomes by O1, O2, . . . , and the sample space by S. Thus, in set-theory notation,
S = {O1, O2,. ..}
The PROBABILITY of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.
3 approaches to calculate probability are :-
1). A priori approach (prior information )
It is the Probability calculated by logically examining existing information. A priori probability can most easily be described as making a conclusion based upon deductive reasoning rather than research or calculation. The largest drawback to this method of defining probabilities is that it can only be applied to a finite set of events.
For 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).
One disadvantage of A priori probability is that it applies only to finite collections of events.
2). Empirical approach(Information collected)
A form of probability that is based on some event occurring, which is calculated using collected empirical evidence. An empirical probability is closely related to the relative frequency in a given probability distribution.
An example for empirical method would be the study of capital asset pricing model which is often used to estimate a companies weighted average cost of capital.
3). Subjective approach (judgemental)
In the subjective approach, we define probability as the degree of belief that we hold in the occurrence of an event. Thus, judgment is used as the basis for assigning probabilities.
Example: Stock Price
What is the probability for a particular stock to go up tomorrow? Again, this “experiment” can’t be repeated, and we can’t apply the relative-frequency approach. Sophisticated models (that rely on past data) are often used to make such predictions, as blindly following
ill-founded judgments is often dangerous.
3 methods of probability
Three methods of Estimation which are as follows -:
1 1) Ratio Change ( varies from 0 to 1 eg.1/10)
2 2) Percentage Change ( varies from 0 to 100 e.g. 50% )
3 3) Probability ( varies from 0 to 1 e.g. 0.1)
RATIO CHANCE - A ratio represents, for every amount of one thing, how much there is of another thing.
Example 3/7
PERCENTAGE CHANGE - In percentage, it’s all about "x" out of 100. Percentage is reported as a ratio but the sum adds up to 100.
Example 40% (40 out of 100)
PROBABILITY – Probability is the chance that something will happen - how likely it is that some event will happen. It is the measure of uncertainty of an event.
1 >= P(E) >= 0
Sometimes you can measure a probability with a number: "10% chance of rain", or you can use words such as impossible, unlikely, possible, even chance, likely and certain.
Example: "It is unlikely to rain tomorrow".
Sometimes you can measure a probability with a number: "10% chance of rain", or you can use words such as impossible, unlikely, possible, even chance, likely and certain.
Example: "It is unlikely to rain tomorrow".
The set of all possible outcomes of an experiment is called the sample space.
We will denote the outcomes by O1, O2, . . . , and the sample space by S. Thus, in set-theory notation,
S = {O1, O2,. ..}
Example:- when events A & B are part of a sample space S. This sample space is represented as a set as in the diagram below :
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 – A schematic diagram used in logic theory to depict collection of sets and represents their relationships, It is an illustration of the relationships between and among sets, group of objects that share something in common. Generally, they are used to depict set intersections and unions. The diagram uses circles and rectangles where circles represents sets with the position and overlap of the circles indicating the relationships between the sets.
3 Event Properties -:
1 1) Mutually Exclusive Events
2 2) Independent Events
3 3) Dependent Events
MUTUALLY EXCLUSIVE EVENTS – Two events that have NO outcomes in common are called Mutually Exclusive. These are the events that cannot occur at the same time.
For Eg: Recording two separate roles of one die are mutually exclusive events. Whatever number the dice displays on its first role will have no impact on what number is rolled the second time.
INDEPENDENT EVENTS – When two events are said to be independent of each other, this means that the probability that one event occurs in no way affects the probability of the other event occurring.
For Eg : You roll a die and flipped a coin. The probability of getting any number face on the die in no way influences the probability of getting a head or a tail on the coin.
DEPENDENT EVENTS – When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event.
For Eg : If you were to draw two cards from a deck of 52 cards. If in your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is changed completely because you drew an ace the first time.
3 Laws of Probability -:
1 1) Addition Law
2 2) Conditional Law
3 3) Multiplicative Law
ADDITIONAL LAW
If two events A & B are mutually exclusive then,
P(AUB) = P(A)+P(B)
This is the simplified version of the additional law. However, when A & B are not mutually exclusive, A∩B = ∅, it can be shown that a more general law applies:
P(AUB) = P(A) +P(B) – P(A∩B)
CONDITIONAL LAW
When the probability of occurrence of an event, say B, depends on the whether or not the other event, say A, has occurred, it is called conditional probability. The probability of B occurring is conditional on the occurrence or otherwise of A.
The conditional probability of B occurring given that event A has occurred is written P(B/A).
P(B/A) = P(A∩B)/P(A)
MULTIPLICATION LAW
If A & B are independent events then,
P(A∩B) = P(A)P(B)
In words, the probability of independent events A & B occurring is the product of the probabilities of the events occurring separately.
BAYES THEOREM – Bayes Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics and inductive logic.
- · An Extension of the Conditional Probability Formula
- · Usually used to find the cause of a given experimental result
- · P(Ai|B)=P(B|Ai)P(Ai)/∑j=ik(B|Aj)P(Aj).
This concluded the day’s sessions which helped me get rid of some myths related to probability and at the same time developed a greater sense of understanding of the topic.
written by:- Pankaj Ruplani
Group Members :- Praloy Kumar Saha
Navdeep Singh
Navneet Singh
Shyam Pandulle
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