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Theorems And Conditional Probability

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- 1. 1.3 Elementary Theoremsand Conditional Probability<br />
- 2. Theorem 1,2<br />Generalization of third axiom of probability<br />Theorem 1: If A1, A2,….,Anare mutually exclusive events in a sample space, then<br />P(A1 A2….An) = P(A1) + P(A2) + …+ P(An).<br />Rule for calculating probability of an event<br />Theorem 2: If A is an event in the finite sample space S, then P(A) equals the sum of the probabilities of the individual outcome comprising A. <br />
- 3. Theorem 3<br />Proof: If E1, E2,……Enbe the n outcomes comprising event A, then A = E1E2 …… En. Since the E’s areindividual outcomesthey are mutually exclusive, and by Theorem 1, we have <br /> P(A) = P(E1E2 …… En)<br /> = P(E1) + P(E2) + …+ P(En).<br />General addition rule for probability<br />Theorem 3: If A and B are any events in S, then<br /> P(AB) = P(A) + P(B) – P(AB).<br />
- 4. Theorem 4<br />Note: When A and B are mutually exclusive so that P(AB) = 0, Theorem 3 reduces to the third axiom of probability therefore the third axiom of probability also called the special addition rule<br />Probability rule of the complements<br />Theorem 4: If A is any event in S, then <br />P(<br />) = 1 – P(A).<br />
- 5. Proof<br />are mutually exclusive by<br />Proof: Since A and<br />definition and A<br />= S. Hence we have<br />) = P(S) = 1.<br />P(A) + P(<br />) = P (A<br />P(<br /> ) = 1 – P(<br />) = 1 – P( S ) = 0.<br />If A B then P(BA) = P(B) - P(A) <br />P(A B) = P(A) + P(B) - 2 P(AB) <br />
- 6. Conditional Probability<br />If we ask for the probability of an event then it is meaningful only if we mention about the sample space.<br />When we use the symbol P(A) for probability of A, we really mean the probability of A with respect to some sample space S. <br />Since there are problems in which we are interested in probabilities of A with respect to more sample spaces than one, the notation P(A|S) is used to make it clear that we are referring to a particular sample space S.<br />
- 7. Conditional Probability<br />P(A|S) conditional probability of A relative to S.<br />Conditional probability: If A and B are any events in S and P(B) 0, the conditional probability of A given B is <br />P (A|B) is the probability that event A occurs once<br />event B has occurred<br />
- 8. Conditional Probability (cont’d)<br />Reduced Sample Space<br />A B<br />S<br />B<br />A<br />P(A|B) measures the relative probability of A with<br />respect to the reduced sample space B<br />
- 9. Conditional Probability (cont’d)<br />If A and B are any two events in the sample space S, Then the event A is independent of the event B if and only if <br />P(A|B) = P(A)<br />i.e. occurrence of B does not influence the occurrence of A. <br />But B is independent of A whenever A is independent of B. <br /> A and B are independent events if and only if <br />either P(A|B) = P(A)or P(B|A) = P(B)<br />
- 10. Conditional Probability<br />General multiplication rule of probability<br />Theorem 5: If A and B are any events in S, then<br />P(AB) = P(A)· P(B|A) if P(A)0<br />= P(B)· P(A|B) if P(B)0<br />Special product rule of probability<br />Theorem 6: Two events A and B are independent events if and only if<br /> P(AB) = P(A)· P(B)<br />
- 11. The mutually exclusive events are not independent unless one of them has zero probability.<br />If an event A is independent of itself then P(A) = 0 or P(A) = 1<br />If the events A and B are independent, then so are events and B, events A and and events and <br /> .<br />
- 12. Bayes’ Theorem<br />Let S be a sample space and B1, B2,….Bnbe mutually exclusive events such that <br /> S = B1B2 …… Bn<br />and A be an event in the sample space S. Then<br /> A = AS = A(B1B2 …… Bn)<br /> = (A B1) (A B2) ……. (A Bn).<br />Since all A Bi ’s are mutually exclusive events <br /> P(A)=P(AB1) + P(AB2) +……. + P(ABn). <br />
- 13. Bayes’ Theorem<br />or<br />But from multiplication rule for probability<br />P(ABi) = P(Bi)·P(A|Bi), for i = 1, 2, …, n<br />hence we have<br />
- 14. Bayes’ Theorem<br />Rule of elimination or rule of total probability<br />Theorem 7 : Let A be an event in a sample space S and if B1, B2,……Bn are mutually exclusive events such that S = B1B2 …… Bnand P (Bi) 0 for i = 1, 2, …, n, then<br />
- 15. Bayes’ Theorem<br />To visualize this result, we have to construct a tree <br /> diagram where the probability of the final outcome is<br /> given by the sum of the products of the probabilities <br /> corresponding to each branch of the tree.<br />P(A|B1)<br />B1<br />A<br />B2<br />P(A|B2)<br />P(B1)<br />A<br />Figure: Tree diagram for rule of elimination<br />P(B2)<br />P(Bn)<br />P(A|Bn)<br />A<br />Bn<br />
- 16. Bayes’ Theorem (cont’d)<br />from the definitionof <br />conditional probability<br />but according to multiplication rule of probability, we have<br />P (Bk A) = P(Bk)·P(A|Bk).<br />Hence, we have<br />
- 17. Bayes’ Theorem<br />Using rule of total probability, we have following result.<br />Bayes’ Theorem<br />Theorem 8: Let A be an event in a sample space S and if B1, B2,……Bn are mutually exclusive events such that S = B1B2 …… Bn and P (Bi) 0 for i = 1, 2, …, n, then<br />for k = 1, 2,….., n.<br />
- 18. Bayes’ Theorem<br />This theorem provides a formula for finding the probability that the “effect” A was “caused” by the event Bk.<br />Note: The expression in the numerator is the probability of reaching A via the kth branch of the tree and the expression in the denominator is the sum of the probabilities of reaching A via the n branches of the tree.<br />

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