Summary

The set of events E_{1}, E_{2}, E_{3}, ..., E_{n} represents a partition of sample space S, if:

â€¢ E_{i} âˆ© E_{j} = âˆ…, i â‰ j , i, j = 1,2,3,...,n

â€¢ E_{1} âˆ© E_{2} âˆ© E_{3} âˆ©...âˆ© E* _{n}* = S P(E

**Theorem of Total Probability:**

Let {E_{1}, E_{2}, ... E* _{n}*} be a partition of sample space S, and suppose that each of the events E

P(A) = P(E

= âˆ‘

**Proof:**

{E_{1}, E_{2}, ..., E* _{n}*} is a partition of sample space S.

âˆ´ S = E_{1}âˆ© E_{2}âˆ© ... âˆ© E_{n}

E* _{i}*âˆªE

P (E* _{i}*) > 0 âˆ€

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

= A âˆ© (E_{1}âˆª E_{2}âˆª ... âˆª E* _{n}*) [âˆµ S = E

= (A âˆ©E_{1}) âˆª(A âˆ©E_{2}) âˆª ... âˆª(A âˆ©E* _{n}*)

A âˆ©E* _{i}*is a subset of E

Given, E* _{i}* and E

âˆ´ A âˆ©E* _{i}* and A âˆ©E

Thus, P(A) = P[(A âˆ©E_{1}) âˆª(A âˆ©E_{2}) âˆª ... âˆª(A âˆ©E* _{n}*)]

= P(A âˆ©E_{1}) + P(A âˆ©E_{2}) + ... + P(A âˆ©E* _{n}*)

By the multiplication rule of probability,

P(A) = P(E_{1})P(A|E_{1}) + P(E_{2})P(A|E_{2}) + ... + P(E* _{n}*)P(A|E

P(A) = âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j})

**Bayes' Theorem:** If E_{1}, E_{2}, ..., E* _{n}* are n non empty events that constitute a partition of sample space S, i.e. E

P(E_{i}|A) = P(E_{i})P(A|E_{i}) / âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j}) âˆ€ i = 1,2,3,......,n

**Proof:**

Given, E_{1}, E_{2}, ..., E* _{n}* is a partition of sample space S.

â‡’ E_{1}, E_{2}, ..., E* _{n}* are pair-wise disjoint

E_{1}âˆª E_{2}âˆª... âˆª E* _{n}* = S

By the formula of conditional probability,

P(E_{i}|A) = P(Aâˆ©E_{i})/P(A)

= P(E_{i})P(A|E_{i}) / P(A) [âˆµ P(Aâˆ©E_{i}) = P(E_{i})P(A|E_{i})]

= P(E_{i})P(A|E_{i}) / âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j}) [âˆµ P(A) = âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j})]

P(E_{i}|A) = P(E_{i})P(A|E_{i}) / âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j}) âˆ€ i = 1,2,3,...., n

Hence it is proved.

Sometimes, Bayes' theorem is called the formula for the probability of causes. Events E_{1}, E_{2}, ..., E* _{n}* are called hypotheses. P(E

The conditional probability, P(E* _{i}*|A), is called the posteriori probability of the hypothesis, E

The set of events E_{1}, E_{2}, E_{3}, ..., E_{n} represents a partition of sample space S, if:

â€¢ E_{i} âˆ© E_{j} = âˆ…, i â‰ j , i, j = 1,2,3,...,n

â€¢ E_{1} âˆ© E_{2} âˆ© E_{3} âˆ©...âˆ© E* _{n}* = S P(E

**Theorem of Total Probability:**

Let {E_{1}, E_{2}, ... E* _{n}*} be a partition of sample space S, and suppose that each of the events E

P(A) = P(E

= âˆ‘

**Proof:**

{E_{1}, E_{2}, ..., E* _{n}*} is a partition of sample space S.

âˆ´ S = E_{1}âˆ© E_{2}âˆ© ... âˆ© E_{n}

E* _{i}*âˆªE

P (E* _{i}*) > 0 âˆ€

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

= A âˆ© (E_{1}âˆª E_{2}âˆª ... âˆª E* _{n}*) [âˆµ S = E

= (A âˆ©E_{1}) âˆª(A âˆ©E_{2}) âˆª ... âˆª(A âˆ©E* _{n}*)

A âˆ©E* _{i}*is a subset of E

Given, E* _{i}* and E

âˆ´ A âˆ©E* _{i}* and A âˆ©E

Thus, P(A) = P[(A âˆ©E_{1}) âˆª(A âˆ©E_{2}) âˆª ... âˆª(A âˆ©E* _{n}*)]

= P(A âˆ©E_{1}) + P(A âˆ©E_{2}) + ... + P(A âˆ©E* _{n}*)

By the multiplication rule of probability,

P(A) = P(E_{1})P(A|E_{1}) + P(E_{2})P(A|E_{2}) + ... + P(E* _{n}*)P(A|E

P(A) = âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j})

**Bayes' Theorem:** If E_{1}, E_{2}, ..., E* _{n}* are n non empty events that constitute a partition of sample space S, i.e. E

P(E_{i}|A) = P(E_{i})P(A|E_{i}) / âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j}) âˆ€ i = 1,2,3,......,n

**Proof:**

Given, E_{1}, E_{2}, ..., E* _{n}* is a partition of sample space S.

â‡’ E_{1}, E_{2}, ..., E* _{n}* are pair-wise disjoint

E_{1}âˆª E_{2}âˆª... âˆª E* _{n}* = S

By the formula of conditional probability,

P(E_{i}|A) = P(Aâˆ©E_{i})/P(A)

= P(E_{i})P(A|E_{i}) / P(A) [âˆµ P(Aâˆ©E_{i}) = P(E_{i})P(A|E_{i})]

= P(E_{i})P(A|E_{i}) / âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j}) [âˆµ P(A) = âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j})]

P(E_{i}|A) = P(E_{i})P(A|E_{i}) / âˆ‘_{j=1}^{n} P(E_{j})P(A|E_{j}) âˆ€ i = 1,2,3,...., n

Hence it is proved.

Sometimes, Bayes' theorem is called the formula for the probability of causes. Events E_{1}, E_{2}, ..., E* _{n}* are called hypotheses. P(E

The conditional probability, P(E* _{i}*|A), is called the posteriori probability of the hypothesis, E