Notes On Bayes' Theorem - CBSE Class 12 Maths

The set of events E1, E2, E3, ..., En represents a partition of sample space S, if:

Ei ∩ Ej = ∅, i ≠ j , i, j = 1,2,3,...,n

E1 E2 E3 ... En = S P(Ei) > 0 ∀ 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)


= ∑j=1n P(Ej)P(A|Ej)

Proof:

{E1, E2, ..., En} is a partition of sample space S.

∴ S = E1 E2 ... En

EiEj = f, ij and i, j = 1, 2, 3, ..., n

P (Ei) > 0 ∀ i = 1,2, 3, ..., n

For any event A associated with S, A = A S

= A (E1 E2 ... En) [∵ S = E1 E2 ... En]

= (A E1) (A E2) ... (A En)

A Eiis a subset of Eiand A Ejis a subset of Ej.

Given, Ei and Ej are disjoint for ij.

∴ A Ei and A Ej are also disjoint ∀ij and i, j = 1, 2, 3, ..., n.

Thus, P(A) = P[(A E1) (A E2) ... (A En)]

= P(A E1) + P(A E2) + ... + P(A 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) = 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 E1E2...En = S, and A is any event of non-zero probability, then


P(Ei|A) = P(Ei)P(A|Ei) / j=1n P(Ej)P(A|Ej) i = 1,2,3,......,n

Proof:

Given, E1, E2, ..., En is a partition of sample space S.

E1, E2, ..., En are pair-wise disjoint

E1 E2... En = S

By the formula of conditional probability,

P(Ei|A) = P(A∩Ei)/P(A)

= P(Ei)P(A|Ei) / P(A) [∵ P(A∩Ei) = P(Ei)P(A|Ei)]

= P(Ei)P(A|Ei) / j=1n P(Ej)P(A|Ej) [∵ P(A) = j=1n P(Ej)P(A|Ej)]

P(Ei|A) = P(Ei)P(A|Ei) / j=1n P(Ej)P(A|Ej) 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:

Ei ∩ Ej = ∅, i ≠ j , i, j = 1,2,3,...,n

E1 E2 E3 ... En = S P(Ei) > 0 ∀ 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)


= ∑j=1n P(Ej)P(A|Ej)

Proof:

{E1, E2, ..., En} is a partition of sample space S.

∴ S = E1 E2 ... En

EiEj = f, ij and i, j = 1, 2, 3, ..., n

P (Ei) > 0 ∀ i = 1,2, 3, ..., n

For any event A associated with S, A = A S

= A (E1 E2 ... En) [∵ S = E1 E2 ... En]

= (A E1) (A E2) ... (A En)

A Eiis a subset of Eiand A Ejis a subset of Ej.

Given, Ei and Ej are disjoint for ij.

∴ A Ei and A Ej are also disjoint ∀ij and i, j = 1, 2, 3, ..., n.

Thus, P(A) = P[(A E1) (A E2) ... (A En)]

= P(A E1) + P(A E2) + ... + P(A 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) = 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 E1E2...En = S, and A is any event of non-zero probability, then


P(Ei|A) = P(Ei)P(A|Ei) / j=1n P(Ej)P(A|Ej) i = 1,2,3,......,n

Proof:

Given, E1, E2, ..., En is a partition of sample space S.

E1, E2, ..., En are pair-wise disjoint

E1 E2... En = S

By the formula of conditional probability,

P(Ei|A) = P(A∩Ei)/P(A)

= P(Ei)P(A|Ei) / P(A) [∵ P(A∩Ei) = P(Ei)P(A|Ei)]

= P(Ei)P(A|Ei) / j=1n P(Ej)P(A|Ej) [∵ P(A) = j=1n P(Ej)P(A|Ej)]

P(Ei|A) = P(Ei)P(A|Ei) / j=1n P(Ej)P(A|Ej) 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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