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Home , Complementary event, Elementary event, Sample space

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PROBABILITY

Let us first see some of the basic terminologies and concepts used in .

EXPERIMENT- An Experiment is an action or operation which can produce some well-defined result/(s). There are two types of experiment:- (1) Deterministic Experiment (2) Indeterministic/Random/Probabilistic/Stochasticuksinghmaths.com Experiment

Deterministic Experiment:- Those experiments which when repeated under identical condition always gives the same outcome/ result, no matter where it is performed and no matter how many times it is performed, are called Deterministic experiment.

Those experiments which when repeated under Indeterministic/Random/ Probabilistic Experiment:- identical conditions do not always give the same output/result, is known as Indeterministic/Random/ Probabilistic/Stochastic experiment.

Trial An experiment once performed is called Trial. uksinghmaths.com

The of all possible outcomes of of a random experiment is called of that experiment and is generally denoted by S. Space:- For example, In case of tossing a coin, the sample space is given by S={H,T}, where H=Head and T=Tail. Similarly in rolling a die, the sample space is given by S={1,2,3,4,5,6}

Each element of a sample space is called a sample point or an .

Event:- Any subset of a sample space is called an Event. For example, let us consider the case of rolling a die, then sample space is given by, S={1,2,3,4,5,6}. Let E1={1}, E2={2,3}, E3={2,4,6) , E4={1,3,5}, Then E1,E2,E3 & E4 all are events, as each are the subset of S. If the outcome of a random experiment is an element of an event then we say that the event has occurred otherwise not. For example, in rolling a die the outcome is ‘1’, then we say that above events E1 & E4 has occurred but E2 & E3 has not occurred, As 1∈E1&E4. Let S be the sample space of an experiment . since null set(empty set) is subset of every set . Hence,Ø⊆S,so Ø is an event. This event is called impossible event. Since every set is the subset of itself. Hence S⊆S,so S is an event and this event is called sure event.

Equally Two events are called equally likely event if the occurrence of one is Likely Event:- not preferred over the other. For example, In tossing a coin the event of getting a head or a tail is equallyuksinghmaths.com likely. Similarly, in case of rolling a die, the event of getting any one of the six digits is equally likely.

Exclusive Two Events are said to be exclusive, if the occurrence of one Events:- excludes the possibility of the occurrence of other. such events are disjoint events. Thus two events A and B are exclusive if and only if A∩B=Ø

Exhaustive Let S be the sample space of an experiment. Let E1,E2,E3,…En Events:- be events such that, E1∪E2∪E3……∪En=S, Then the set of these events are called exhaustive events. uksinghmaths.com

Let S be the sample space of an experiment. If E1,E2,E3,………En be sets of events such that: Exclusively Exhaustive (1) E1∪E2∪E3……… ∪En=S Events:- (2) Ei∩Ej=Ø (i≠j) are called exclusively exhaustive event.

Complementary Event:- Two events A and B of the sample space S are said to be if, (1) A∪B=S (2) A∩B=Ø The complement of A is generally denoted by A’.

Independent Event:- Two events are said to be independent of each other if the happening of one does not effect the happening of other.

Dependent Event:- Two events are said to be dependent if the occurrence of one affects the occurrence of other.

Probability:- Probability is a concept which numerically measures the degree of uncertainty and therefore degree of certainty of the occurrence of events. Let S be the Sample space associated with some random experiment. Let E ⊆S be any event. Then the Probability of occurrence of E is denoted P(E) and is defined as, P(E)=n(E)/n(S), Where n stands for number of elements. =Numberuksinghmaths.com of outcomes favourable to E/Total number of possible outcomes For example, let us consider the experiment of rolling a die, the sample space is given by, S={1,2,3,4,5,6}. Let E={2,4} be an event then, P(E)=n(E)/n(S)=2/6=1/3.

The probability of impossible event is zero. As, P(Ø)=n(Ø)/n(S)=0/n(S)=0 The probability of sure event is always ‘1’. As, P(S)=n(S)/n(S)=1 The probability of any event lies between o and 1. 0≤P(E)≤1 uksinghmaths.com

Odds in favour and against of an event: If an event A can occur in m ways and can fail in n ways .

odds in favour of the event A is given by Odds m/n=no.of favourable choices/no. of unfavourable choices. in favour =no. of successes/no. of failures. =probability of occurrence of event{m/m+n)}/ probability of not occurrence of event{n/m+n)}. =P(A)/P(A'). For example, odds in favour of rolling a 6 on a fair six sided die is1/5.

Odds against of the event A is given by Odds n/m=no. of unfavourable choices/no. of favourable choices. in favour =no. of failures/no. of successes. =probability of not occurrence of event{n/(m+n)}/ probability of occurrence of event{m/(m+n)}. =P(A')/P(A). For example,odds against of rolling a 6 on a fair six sided die is 5/1. Q. Find the odds in favour of getting exactly two heads when three coins are tossed. Sol.- The sample space of tossing three coins are given by S={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT} Thus,the odds in favour of exactly two head=3/5.

Theorems of probability S Addition Rule of Let S be the sample space associated Probability:- with a random experiment and A & B A B uksinghmaths.comare two events.Then, by n(A)∪n(B)=n(A)+n(B)-n(A∩B) ⇒{n(A)∪n(B)}/n(S)={n(A)+n(B)-n(A∩B)}/n(S) ⇒{n(A∪B)/n(S)}={n(A)/n(S)}+{n(B)/n(S)}- {n(A∩B)}/n(S)} ⇒P(A∪B)=P(A)+P(B)-P(A∩B) s

If A and B are mutually exclusive events then A B A∩B=Ø ⇒P(A∩B)=P(Ø)=0 A & B are mutually exclusive events Thus, P(A∪B)=P(A)+P(B) uksinghmaths.com

For complementary events, A∪A’=S & A∩A’=Ø ⇒P(A∪A’)=P(S) ⇒P(A)+P(A’)=1 ⇒P(A’)=1-P(A)

Let S be the sample space associated with a random experiment and A,B and C be three events associated with it. Then P(A∪B∪C)=P(A)+P(B)+P(C)-P(A∩B)-P(B∩C)-P(C∩A)+P(A∩B∩C)

If A,B & C are mutually exclusive events,Then A∩B=B∩C=C∩A=A∩B∩C=Ø ⇒P(A∩B)=P(B∩C)=P(C∩A)=P(A∩B∩C)=P(Ø)=0 ⇒P(A∪B∪C)=P(A)+P(B)+P(C)

Multiplication Rule of Probability or Let S be the sample space associated with Theorem of compound Probability:- some random experiment and A,B are its two events. Let event B has occurred and B≠Ø . Then, no. of favourable cases of A=n(A∩B),after occurrence of event B. Now,S will not be the sample space for A,here the sample space will be B. No. of elements in sample space of A will be n(B).

P(A | B), means probability of event A given event B has occurred.This is also known as A given even B has already occurred, and is defineduksinghmaths.com as P(A | B)=n(A∩B)/n(B) ={n(A∩B)/n(S)}/{n(B)/n(S)} =P(A∩B)/P(B) ⇒P(A | B)=P(A∩B)/P(B) ⇒P(A∩B)=P(A | B)*P(B) Similarly,P(B | A)=P(A∩B)/P(A) , & P(A∩B)=P(B | A)*P(A) uksinghmaths.com

If A and B are two independent events then, P(A∩B)=P(A)*P(B) Proof- From multiplication theorem, we know that for any two events A and B, we know that P(A∩B)=P(A | B)*P(B)………..(i) But A & B are independent events,So occurrence of B does not efects occurrence of A. Hence,P(A | B)=P(A) Now putting this value in eqn.(i), we have P(A∩B)=P(A)*P(B). If A and B are not independent events, Then P(A∩B)≠P(A)*P(B) If A,B,C are three independent events, Then P(A∩B∩C)=P(A)*P(B)*P(C) Proof- Let B∩C=E Since A,B,C are independent events, therefore A and E are also independent. Since A,E are independent events, then P(A∩E)=P(A)*P(E) =P(A)*P(B∩C) =P(A)*P(B)*P(C) {as, B and C are independent,hence P(B∩C)=P(B)*P(C)} ⇒P(A∩B∩C)=P(A)*P(B)*P(C). Now,uksinghmaths.com mathematical notations for some events: Complementary of A or Negation A or Not A ⇒ ¯A A and B ⇒ A∩B A or B (at least one of A or B) ⇒ A∪B A but not B ⇒A∩¯B Neither A nor B⇒¯A ∩¯B Exactly one of A and B⇒(¯A ∩ B) ∪ (A ∩¯B) All three of A,B and C ⇒ A∩B∩C At least one of A,B or C ⇒ A∪B∪C Exactly two of A,B and C⇒ (¯A∩B∩C)∪(A∩¯B∩C)∪(A∩B∩¯C) uksinghmaths.com

Now, Sample Space of some Well known Random experiments. Sample space of Tossing a fair coin once is given by S={H,T}

H T

Sample space of tossing two unbiased coins simultaneously or tossing a single coin two times is given by S={HH,HT,TH,TT}

Sample space of tossing three unbiased coins simultaneously or tossing a single coin three times is given by S={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}

Sample space of tossing four unbiased coins simultaneously or tossing a single coin four times is given by S={HHHH,HHHT,HTHH,THHH,HHTH,HHTT,HTTH,TTHH,THHT,HTHT, THTH,TTTH,TTHT,THTT,HTTT,TTTT}

Sample space of rolling a die is given by S={1,2,3,4,5,6} uksinghmaths.com Sample space of rolling two dice simultaneously is given by 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)} uksinghmaths.com

Sample space of a well shufed pack of cards has 52 no. of cards, so the sample space has 52 elements and their individual composition is given below:

A COMPLETE DECK OF CARDS consists of 52 cards arranged in 4 suite each suite contains 13 cards known as Club, Spade, Heart and Diamond. The two suite of Club and Spade (i.e. 26 cards) are Black in colour and two suite of Heart and diamond (i.e. 26 cards) of are red in colour. Each of four suite of cards consists of 13 cards-each having 3 face cards namely Jack,Queen & King and one ace card each & rest 9 cards numbered from 2 to 10. Thus there are 4 kings, 4 queens,4 jacks card of different types in a pack of cards. uksinghmaths.comThus there are total 12 face cards. uksinghmaths.com

PROBABILITY THEORY HISTORICAL BACKGROUND

The theory of probability developed as a result of gamblers disputes & in the race of ways to cheat in these games. had its origin in 16th century when Italian physicist and mathematician J.Cardan(1501 – 1576) ,wrote the first book on the subject,The book on Games of Chance,which introduced the idea of probability but no general theory was developed at this time. Because of the inherent appeal of games of chance, probability theory soon became popular, and the subject developed rapidly during the 18th century. Since its inception, the study of probability has attracted the attention of great mathematicians. James Bernoulli(1654-1705), Abraham de Moivre(1667-1754), and pierre Simon Laplace are among those who made significant contribution to this field. Laplace’s Theorie Analytique des Probabilités,1812 is considered to be the greatest contribution by single person to the theory of probability. The problem posed by de Méré led to an exchange of letters between Pascal and Fermat in which the fundamental principles of probability theory were formulated for the first time. Before Laplace, probability theory was solely concerned with developing a mathematical analysis of games of chance. Laplace applied probabilistic ideas to diverse areas like for many scientific and practical problems. Today probabilityuksinghmaths.com and its branch of Mathematical is used in many areas such as Biology,Economics,Genetics,Physics, Sociology, Psychology & Engineering etc. Many persons have contributed to the theory since Laplace's time; among the most important are Chebyshev, Markov, von Mises, and Kolmogorov.