This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 2 Conditional Probability • Suppose that we have partial knowledge of the result of a trial of the experiment — we know that event A occurred on the trial — but we do not know which outcome in A occurred; only that the outcome is some member of event A • Did the event B occur? • We cannot say for sure: All we know is that the actual outcome x is a member of A, and all we can say is that B occurred if and only if x is a member of B as well , that is, B occurred if x ∈ AB, and B did not occur if x ∈ AB c • Event A is known to have occurred; but we don’t know which outcome in A occurred • Event B has occurred if the actual outcome belongs to AB • Event B has not occurred (i.e B c has occurred) if the actual outcome belongs to AB c • Since we cannot say for sure whether B occurred or not, we would like a probabilistic description of the situation • P(B) and P(B c ) should be updated in view of the partial knowledge that the event A occurred on this particular trial of the experiment • Special cases: • If A ⊂ B, then x ∈ A ⇒ x ∈ B; hence B occurs whenever A does. We should update the probability of B from P(B) to 1 • If AB = ø, x ∈ A ⇒ x ∉ B; hence B cannot occur when A does. We should update the probability of B from P(B) to 0 • More generally, how should we update P(B) in light of the partial knowledge that event A occurred? • We need nomenclature and notation to distinguish between the probability that we originally assigned to the event B and the updated probability of the event B • P(B), the number originally assigned as the probability of B, is called the unconditional or a priori probability of B • The updated probability of B is called the conditional probability of B given A (or the probability of B given A) and is denoted by P(B  A) • Assume P(A) > 0 • P(B  A), the conditional probability of B given A, is P(B  A) = P(B ∩ A) P(A) = P(BA) P(A) • Similarly, P(B c  A) = P(B c A) P(A) • A is the conditioning event • P(B  A) = P(BA) P(A) • P(B c  A) = P(B c A) P(A) • These definitions also work in the special cases considered above. • Example: If A ⊂ B ⇒ BA = A, and B c A = ø A B • Hence, P(B  A) = 1; P(B c  A) = 0 © 1997 by Dilip V. Sarwate. All rights reserved. No part of this manuscript may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the author. Chapter 2 Conditional Probability 17 • Example: If BA = ø, then P(B  A) = 0. Also B c A = A ⇒ P(B c  A) = 1 A B • Example: Two fair dice are rolled. What is the (conditional) probability that the sum of the two faces is 6 given that the two dice are showing different faces?...
View
Full
Document
This test prep was uploaded on 09/28/2007 for the course BTRY 4080 taught by Professor Schwager during the Fall '06 term at Cornell.
 Fall '06
 SCHWAGER
 Conditional Probability, Probability, Probability theory, University of Illinois

Click to edit the document details