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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?...
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 Fall '06
 SCHWAGER
 Conditional Probability, Probability, Probability theory, University of Illinois

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