Notes On Bayes' Theorem - CBSE Class 12 Maths
The set of events E1, E2, E3, ..., En represents a partition of sample space S, if: â€¢ Ei âˆ© Ej = âˆ…, i â‰  j , i, j = 1,2,3,...,n â€¢ E1 âˆ© E2 âˆ© E3 âˆ©...âˆ© En = S P(Ei) > 0 âˆ€ i = 1,2,3,...,n Theorem of Total Probability: Let {E1, E2, ... En} be a partition of sample space S, and suppose that each of the events E1, E2, ... En has non-zero probability of occurrence. Let A be any event associatedwith S.Then P(A) = P(E1)P(A|E1) + P(E2)P(A|E2) + ... + P(En)P(A|En) = âˆ‘j=1n P(Ej)P(A|Ej) Proof: {E1, E2, ..., En} is a partition of sample space S. âˆ´ S = E1âˆ© E2âˆ© ... âˆ© En EiâˆªEj = f, i â‰  j and i, j = 1, 2, 3, ..., n P (Ei) > 0 âˆ€ i = 1,2, 3, ..., n For any event A associated with S, A = A âˆ© S = A âˆ© (E1âˆª E2âˆª ... âˆª En) [âˆµ S = E1âˆª E2âˆª ... âˆª En] = (A âˆ©E1) âˆª(A âˆ©E2) âˆª ... âˆª(A âˆ©En) A âˆ©Eiis a subset of Eiand A âˆ©Ejis a subset of Ej. Given, Ei and Ej are disjoint for i â‰  j. âˆ´ A âˆ©Ei and A âˆ©Ej are also disjoint âˆ€i â‰  j and i, j = 1, 2, 3, ..., n. Thus, P(A) = P[(A âˆ©E1) âˆª(A âˆ©E2) âˆª ... âˆª(A âˆ©En)] = P(A âˆ©E1) + P(A âˆ©E2) + ... + P(A âˆ©En) By the multiplication rule of probability, P(A) = P(E1)P(A|E1) + P(E2)P(A|E2) + ... + P(En)P(A|En) P(A) = âˆ‘j=1n P(Ej)P(A|Ej) Bayes' Theorem: If E1, E2, ..., En are n non empty events that constitute a partition of sample space S, i.e. E1, E2, ..., En are pair-wise disjoint and E1âˆªE2âˆª...âˆªEn = S, and A is any event of non-zero probability, then P(Ei|A) = P(Ei)P(A|Ei) / âˆ‘j=1n P(Ej)P(A|Ej) âˆ€ i = 1,2,3,......,n Proof: Given, E1, E2, ..., En is a partition of sample space S. â‡’ E1, E2, ..., En are pair-wise disjoint E1âˆª E2âˆª... âˆª En = S By the formula of conditional probability, P(Ei|A) = P(Aâˆ©Ei)/P(A) = P(Ei)P(A|Ei) / P(A) [âˆµ P(Aâˆ©Ei) = P(Ei)P(A|Ei)] = P(Ei)P(A|Ei) / âˆ‘j=1n P(Ej)P(A|Ej) [âˆµ P(A) = âˆ‘j=1n P(Ej)P(A|Ej)] P(Ei|A) = P(Ei)P(A|Ei) / âˆ‘j=1n P(Ej)P(A|Ej) âˆ€ i = 1,2,3,...., n Hence it is proved. Sometimes, Bayes' theorem is called the formula for the probability of causes. Events E1, E2, ..., En are called hypotheses. P(Ei) is called the priori probability of the hypothesis, Ei. The conditional probability, P(Ei|A), is called the posteriori probability of the hypothesis, Ei.

#### Summary

The set of events E1, E2, E3, ..., En represents a partition of sample space S, if: â€¢ Ei âˆ© Ej = âˆ…, i â‰  j , i, j = 1,2,3,...,n â€¢ E1 âˆ© E2 âˆ© E3 âˆ©...âˆ© En = S P(Ei) > 0 âˆ€ i = 1,2,3,...,n Theorem of Total Probability: Let {E1, E2, ... En} be a partition of sample space S, and suppose that each of the events E1, E2, ... En has non-zero probability of occurrence. Let A be any event associatedwith S.Then P(A) = P(E1)P(A|E1) + P(E2)P(A|E2) + ... + P(En)P(A|En) = âˆ‘j=1n P(Ej)P(A|Ej) Proof: {E1, E2, ..., En} is a partition of sample space S. âˆ´ S = E1âˆ© E2âˆ© ... âˆ© En EiâˆªEj = f, i â‰  j and i, j = 1, 2, 3, ..., n P (Ei) > 0 âˆ€ i = 1,2, 3, ..., n For any event A associated with S, A = A âˆ© S = A âˆ© (E1âˆª E2âˆª ... âˆª En) [âˆµ S = E1âˆª E2âˆª ... âˆª En] = (A âˆ©E1) âˆª(A âˆ©E2) âˆª ... âˆª(A âˆ©En) A âˆ©Eiis a subset of Eiand A âˆ©Ejis a subset of Ej. Given, Ei and Ej are disjoint for i â‰  j. âˆ´ A âˆ©Ei and A âˆ©Ej are also disjoint âˆ€i â‰  j and i, j = 1, 2, 3, ..., n. Thus, P(A) = P[(A âˆ©E1) âˆª(A âˆ©E2) âˆª ... âˆª(A âˆ©En)] = P(A âˆ©E1) + P(A âˆ©E2) + ... + P(A âˆ©En) By the multiplication rule of probability, P(A) = P(E1)P(A|E1) + P(E2)P(A|E2) + ... + P(En)P(A|En) P(A) = âˆ‘j=1n P(Ej)P(A|Ej) Bayes' Theorem: If E1, E2, ..., En are n non empty events that constitute a partition of sample space S, i.e. E1, E2, ..., En are pair-wise disjoint and E1âˆªE2âˆª...âˆªEn = S, and A is any event of non-zero probability, then P(Ei|A) = P(Ei)P(A|Ei) / âˆ‘j=1n P(Ej)P(A|Ej) âˆ€ i = 1,2,3,......,n Proof: Given, E1, E2, ..., En is a partition of sample space S. â‡’ E1, E2, ..., En are pair-wise disjoint E1âˆª E2âˆª... âˆª En = S By the formula of conditional probability, P(Ei|A) = P(Aâˆ©Ei)/P(A) = P(Ei)P(A|Ei) / P(A) [âˆµ P(Aâˆ©Ei) = P(Ei)P(A|Ei)] = P(Ei)P(A|Ei) / âˆ‘j=1n P(Ej)P(A|Ej) [âˆµ P(A) = âˆ‘j=1n P(Ej)P(A|Ej)] P(Ei|A) = P(Ei)P(A|Ei) / âˆ‘j=1n P(Ej)P(A|Ej) âˆ€ i = 1,2,3,...., n Hence it is proved. Sometimes, Bayes' theorem is called the formula for the probability of causes. Events E1, E2, ..., En are called hypotheses. P(Ei) is called the priori probability of the hypothesis, Ei. The conditional probability, P(Ei|A), is called the posteriori probability of the hypothesis, Ei.
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