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Probability Theory Umpt001 uksinghmaths.com PROBABILITY Let us first see some of the basic terminologies and concepts used in probability. EXPERIMENT- An Experiment is an action or operation which can produce some well-defined result/outcome(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 set of all possible outcomes of of a random experiment is called Sample Sample space 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 elementary event. 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 complementary event 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 odds 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 venn diagram 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 conditional probability 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).
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