Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

1,377 views

1,271 views

1,271 views

Published on

Theorems And Conditional Probability

No Downloads

Total views

1,377

On SlideShare

0

From Embeds

0

Number of Embeds

3

Shares

0

Downloads

0

Comments

0

Likes

2

No embeds

No notes for slide

- 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 />

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment